Translating a planar object to maximize point containment Exact and approximation algorithm
DMAX中英文翻译大全对照表
Absolute Mode Transform Type-in绝对坐标方式变换输入Absolute/Relative Snap Toggle Mode绝对/相对捕捉开关模式ACIS Options ACIS选项Activate活动；激活Activate All Maps激活所有贴图Activate Grid激活栅格；激活网格Activate Grid Object激活网格对象；激活网格物体Activate Home Grid激活主栅格；激活主网格ActiveShade实时渲染视图；着色；自动着色ActiveShade(Scanline)着色(扫描线) ActiveShade Floater自动着色面板；交互渲染浮动窗口ActiveShade Viewport自动着色视图Adaptive适配；自动适配；自适应Adaptive Cubic立方适配Adaptive Degradation自动降级Adaptive Degradation Toggle降级显示开关Adaptive Linear线性适配Adaptive Path自适应路径Adaptive Path Steps适配路径步幅；路径步幅自动适配Adaptive Perspective Grid Toggle适配透视网格开关Add as Proxy加为替身Add Cross Section增加交叉选择Adopt the File's Unit Scale采用文件单位尺度Advanced Surface Approx高级表面近似；高级表面精度控制Advanced Surface Approximation高级表面近似；高级表面精度控制Adv. Lighting高级照明Affect Diffuse Toggle影响漫反射开关Affect Neighbors影响相邻Affect Region影响区域Affect Region Modifier影响区域编辑器；影响区域修改器Affect Specular Toggle影响镜面反射开关AI Export输出Adobe Illustrator(＊.AI)文件AI Import输入Adobe Illustrator(＊.AI)文件Align对齐Align Camera对齐摄像机Align Grid to View对齐网格到视图Align Normals对齐法线Align Orientation对齐方向Align Position对齐位置(相对当前坐标系)Align Selection对齐选择Align to Cursor对齐到指针Allow Dual Plane Support允许双面支持All Class ID全部类别All Commands所有命令All Edge Midpoints全部边界中点；所有边界中心All Face Centers全部三角面中心；所有面中心All Faces所有面All Keys全部关键帧All Tangents全部切线All Transform Keys全部变换关键帧Along Edges沿边缘Along Vertex Normals沿顶点法线Along Visible Edges沿可见的边Alphabetical按字母顺序Always总是Ambient阴影色；环境反射光Ambient Only只是环境光；阴影区Ambient Only Toggle只是环境光标记American Elm美国榆树Amount数量Amplitude振幅；幅度Analyze World分析世界Anchor锚Angle角度;角度值Angle Snap Toggle角度捕捉开关Animate动画Animated动画Animated Camera/Light Settings摄像机/灯光动画设置Animated Mesh动画网格Animated Object动画物体Animated Objects运动物体；动画物体；动画对象Animated Tracks动画轨迹Animated Tracks Only仅动画轨迹Animation动画Animation Mode Toggle动画模式开关Animation Offset动画偏移Animation Offset Keying动画偏移关键帧Animation Tools动画工具Appearance Preferences外观选项Apply Atmospherics指定大气Apply-Ease Curve指定减缓曲线Apply Inverse Kinematics指定反向运动Apply Mapping指定贴图坐标Apply-Multiplier Curve指定增强曲线Apply To指定到；应用到Apply to All Duplicates指定到全部复本Arc弧；圆弧Arc Rotate弧形旋转；旋转视图；圆形旋转Arc Rotate Selected弧形旋转于所有物体；圆形旋转选择物；选择对象的中心旋转视图Arc Rotate SubObject弧形旋转于次物体；选择次对象的中心旋转视图Arc ShapeArc Subdivision弧细分；圆弧细分Archive文件归档Area区域1Array阵列Array Dimensions阵列尺寸；阵列维数Array Transformation阵列变换ASCII Export输出ASCII文件Aspect Ratio纵横比Asset Browser资源浏览器Assign指定Assign Controller分配控制器Assign Float Controller分配浮动控制器Assign Position Controller赋予控制器Assign Random Colors随机指定颜色Assigned Controllers指定控制器At All Vertices在所有的顶点上At Distinct Points在特殊的点上At Face Centers 在面的中心At Point在点上Atmosphere氛围；大气层；大气，空气；环境Atmospheres氛围Attach连接；结合；附加Attach Modifier结合修改器Attach Multiple多项结合控制；多重连接Attach To连接到Attach To RigidBody Modifier连接到刚性体编辑器Attachment连接；附件Attachment Constraint连接约束Attenuation衰减AudioClip音频剪切板AudioFloat浮动音频Audio Position Controller音频位置控制器AudioPosition音频位置Audio Rotation Controller音频旋转控制器AudioRotation音频旋转Audio Scale Controller音频缩放控制器AudioScale音频缩放；声音缩放Auto自动Auto Align Curve Starts自动对齐曲线起始节点Auto Arrange自动排列Auto Arrange Graph Nodes自动排列节点Auto Expand自动扩展Auto Expand Base Objects自动扩展基本物体Auto Expand Children自动扩展子级Auto Expand Materials自动扩展材质Auto Expand Modifiers自动扩展修改器Auto Expand Selected Only自动扩展仅选择的Auto Expand Transforms自动扩展变换Auto Expand XYZ Components自动扩展坐标组成Auto Key自动关键帧Auto-Rename Merged Material自动重命名合并材质Auto Scroll自动滚屏Auto Select自动选择Auto Select Animated自动选择动画Auto Select Position自动选择位置Auto Select Rotation自动选择旋转Auto Select Scale自动选择缩放Auto Select XYZ Components自动选择坐标组成Auto-Smooth自动光滑AutoGrid自动网格；自动栅格AutoKey Mode Toggle自动关键帧模式开关Automatic自动Automatic Coarseness自动粗糙Automatic Intensity Calculation自动亮度计算Automatic Reinitialization自动重新载入Automatic Reparam.自动重新参数化Automatic Reparamerization自动重新参数化Automatic Update自动更新Axis轴；轴向；坐标轴Axis Constraints轴向约束Axis Scaling轴向比率Back后视图Back Length后面长度Back Segs后面片段数Back View背视图Back Width后面宽度Backface Cull背面忽略显示；背面除去；背景拣出Backface Cull Toggle背景拣出开关Background背景Background Display Toggle背景显示开关Background Image背景图像Background Lock Toggle背景锁定开关Background Texture Size背景纹理尺寸；背景纹理大小Backgrounds背景Backside ID内表面材质号Backup Time One Unit每单位备份时间Banking倾斜Banyan榕树Banyan tree榕树Base基本；基部；基点；基本色；基色Base/Apex基点Base Color基准颜色；基本颜色Base Colors基准颜色Base Curve基本曲线Base Elev基准海拔；基本海拔Base Objects导入基于对象的参数，例如半径、高度和线段的数目；基本物体Base Scale基本比率Base Surface基本表面；基础表面Base To Pivot中心点在底部Bevel Profile轮廓倒角2Bevel Profile Modifier轮廓倒角编辑器；轮廓倒角修改器Bezier贝塞尔曲线Bezier Color贝塞尔颜色Bezier-Corner拐角贝兹点Bezier Float贝塞尔浮动Bezier Lines贝塞尔曲线Bezier or Euler Controller贝塞尔或离合控制器Bezier Position贝塞尔位置Bezier Position Controller贝塞尔位置控制器Bezier Scale贝塞尔比例；贝兹缩放Bezier Scale Controller贝塞尔缩放控制器Bezier-Smooth光滑贝兹点Billboard广告牌Biped步迹；两足Birth诞生；生产Birth Rate再生速度Blast爆炸Blend混合；混合材质；混合度；融合；颜色混合；调配Blend Curve融合曲线Blend Surface融合曲面Blend to Color Above融合到颜色上层；与上面的颜色混合Blizzard暴风雪Blizzard Particle System暴风雪粒子系统Blowup渲染指定区域(必须保持当前视图的长宽比)；区域放大Blue Spruce蓝色云杉Blur模糊Body主体；身体；壶身Body Horizontal身体水平Body Rotation身体旋转Body Vertical身体垂直Bomb爆炸Bomb Space Warp爆炸空间变形Bone骨骼Bone Object骨骼物体；骨骼对象Bone Objects骨骼物体；骨骼对象Bone Options骨骼选项Bone Tools骨骼工具Bones骨骼Bones/Biped骨骼/步迹Bones IK Chain骨骼IK链Bones Objects骨骼物体Boolean布尔运算Boolean Compound Object布尔合成物体Boolean Controller布尔运算控制器Both二者；全部Bottom底；底部；底部绑定物；底视图Bottom View底视图Bounce弹力；反弹；反弹力Bound to Object Pivots绑定到物体轴心Bounding Box边界盒Box方体Box Emitter立方体发射器Box Gizmo方体线框Box Gizmo(Atmospheres)方体线框(氛围)Box Mode Selected被选择的物体模式Box Mode Selected Toggle被选择的物体模式开关Box Selected按选择对象的边界盒渲染；物体长宽比BoxGizmo立方体框；方体线框Break Both行列打断Break Col列打断Break Row行打断Bridge过渡Bright亮度Brightness亮度Bring Selection In加入选择；加入选择集Bubble膨胀；改变截面曲线的形状；气泡；浮起Bubble Motion泡沫运动；气泡运动Bubbles气泡；泡沫；改变截面曲线的形状；膨胀Build Only At Render Time仅在渲染时建立By Material Within Layer按层中的材质Calc Intervals Per Frame计算间隔帧；计算每帧间隔Camera摄像机视图；镜头点；摄像机Camera Point摄像机配合点Camera Point Object摄像机配合点物体CamPoint相机配合点Cancel Align取消对齐Cap封盖；封顶；盖子Cap Closed Entities封闭实体Cap End封底Cap Height顶面高度；顶盖高度Cap Holes封闭孔洞Cap Holes ModifierCap Segments端面片段数Cap Start始端加盖；封闭起端Cap Surface封盖曲面Capping顶盖Capsule囊体；胶囊；胶囊体Capsule Object胶囊体；囊体Case Sensitive区分大小写Cast Shadows投射阴影Center Point Cycle中心点循环Center&Sides中心和边Centered,Specify Spacing居中，指定间距Centimeters厘米C-Ext C型物体；C型延伸体；C型墙C-Extrusion Object C型物体；C型延伸体；C型墙Chamfer倒角；切角Chamfer Curve曲线倒角；切角曲线Chamfer Cylinder倒角圆柱体3Chamfer Cylinder Object倒角圆柱体Chamfer Edge倒角边缘Chamfer Vertex倒角顶点ChamferBox倒角长方体；倒角方体；倒角立方体ChamferBox Object倒角长方体；倒角方体；倒角立方体ChamferCyl倒角圆柱体；倒角柱体Change改变Change Graphics Mode改变图形模式Change Leg StateChange Light Color改变灯光颜色Change to Back Viewport改变到后视图Change to Bottom Viewport改变到底视图Change to Camera Viewport改变到摄像机视图Change to Front View改变到前视图Change to Grid View改变到栅格视图Change to Isometric User View改变到用户轴测视图Change to Left View改变到左视图Change to Perspective User View改变到用户透视视图Change to Right View改变到右视图Change to Shape Viewport改变到二维视图Change to Spot/Directional Light View改变到目标聚光灯/平行光视图Change to Top View改变到顶视图Change to Track View改变到轨迹视图Channel通道Chaos混乱；混乱度Character角色Character Structures角色结构Child孩子Children子级Chop切除；切劈Chord Length弦长；弦长度Circle圆；圆形；圆形区域Circle Shape圆形Circular Region圆形区域Circular Selection Region圆形选择区域Clear清除Clear All清除全部；清除所有的捕捉设置Clear All Smoothing Groups清除全部光滑组Clear Selection清除选择Clear Set Key Mode BufferClear Surface Level清除表面级Click and drag to begin creationprocess单击并拖动，开始创作Clone复制；克隆Clone Method克隆方式；复制方法Close Cols.闭合列Close Loft闭合放样Close Rows闭合行Cloth布；布料Cloth Collection采集布料Cloth Modifier布料编辑器；布料修改器Cloud云Col列Collapse坍塌；塌陷Collapse All全部坍塌；全部折叠Collapse Controller坍塌控制器Collapse Stack坍塌堆栈Collapse To坍塌到；折叠到Color颜色Color by Elevation根据海拔指定颜色；以标高分色Color RGB颜色RGBColor Zone色带Combine合并；联合Combos复合；过滤器组合Combustion燃烧；合并Comet彗星Command Panel命令面板Common Hose Parameters软管共同参数Compare比较Compass指南针；指针Compass Object指针物体Completely replace current scene完全替换当前场景Component组成Composite合成；复合材质；合成贴图；复合Compound Object合成物体Compound Objects合成物体Cone锥体Cone Angle锥体角度Cone Object锥体Configure设置；配置Configure Driver设置驱动Configure OpenGL配置OpenGL显示驱动Configure Paths设置路径Conform包裹Conform Compound Object包裹合成物体Conform Space Warp包裹空间扭曲Connect连接Connect Compound Object连接包裹合成物体Connect Edge连接边界Connect Vertex连接顶点Constant晶体；定常；连续的；连续性；恒定；常量；圆片Constant Cross-Section截面恒定Constant Key Reduction Filtering减少过滤时关键帧不变Constant Velocity匀速Constrain to X约束到X轴Constrain to XY约束到XY轴Constrain to Y约束到Y轴Constrain to Z约束到Z轴Constrained Motion约束运动Constraint约束4Constraints约束Context前后关系；关联菜单Contour轮廓Contours轮廓Contrast对比度Controller控制器；选择用于控制链接对象的关联复制类型Controller Defaults默认控制器Controller Defaults Dialog默认控制器对话框Controller Output控制器输出Controller Properties控制器属性Controller Range控制器范围Controller Range Editor控制器范围编辑器Controller Types控制器类型Controllers控制器Convert blocks to groups转化块为群组Convert Curve转换曲线Convert Curve On Surface在曲面上转换曲线Convert Groups To转化群组到Convert Instances to Blocks转化关联属性为块Convert Selected转换选择；转换当前选择Convert Surface转换曲面Convert To转换到Convert to Edge转换到边Convert to Editable Mesh转换到可编辑网格Convert to Editable Patch转换到可编辑面片Convert to Editable Polygon转换到可编辑多边形Convert to Editable Spline转换到可编辑曲线Convert to Face转换到面Convert to NURBS Surface转换到NURBS曲面Convert To Patch Modifier转换到面片修改器Convert to single objects转化到单一物体Convert to Toolbar转化到工具行；Convert to Vertex转换到顶点Convert units转换单位Cookie Cutter切割；饼切Copies复制数目Copy复制Copy Envelope复制封皮Copy Normal复制法线Corner拐角点Count数量Crawl Time爬行时间；蠕动时间；变动时间Create a Character创建角色Create a Key for all Transforms为所有变换创建关键帧Create a Multicurve Trimmed Surface创建多重修剪表面；创建多重修剪曲面Create a Multisided Blend Surface创建多边的融合表面；创建多边的融合曲面Create a Position Key创建位置关键帧Create a Position Key on X创建X轴位置关键帧Create a Position Key on Y创建Y轴位置关键帧Create a Position Key on Z创建Z轴位置关键帧Create a Rotation Key创建旋转关键帧Create a Rotation Key on X创建X轴的旋转关键帧Create a Rotation Key on Y创建Y轴的旋转关键帧Create a Rotation Key on Z创建Z轴的旋转关键帧Create a Scale Key创建放缩关键帧Create a Scale Key on X创建X轴的放缩关键帧Create a Scale Key on Y创建Y轴的放缩关键帧Create a Scale Key on Z创建Z轴的放缩关键帧Create Blend Curve创建融合曲线Create Blend Surface创建融合表面；创建融合曲面Create Bones System创建骨骼系统Create Cap Surface创建加顶表面；创建加顶曲面Create Chamfer Curve创建倒直角曲线Create Combination创建组合Create Command Mode创建命令模式Create Curves创建曲线Create Curve-Curve创建曲线-曲线Create Curve Point创建曲线点Create CV Curve创建可控曲线；创建控制点曲线Create CV Curve on Surface创建表面CV曲线；创建表面可控曲线Create CV Surface创建CV表面；创建可控曲面Create Defaults创建默认；创建默认值Create Edge创建边Create Explicit Key Position X创建X轴的位置直接关键帧Create Explicit Key Position Y创建Y轴的位置直接关键帧Create Explicit Key Position Z创建Z轴的位置直接关键帧Create Explicit Key Rotation X创建X轴的旋转直接关键帧Create Explicit Key Rotation Y创建Y轴的旋转直接关键帧Create Explicit Key Rotation Z创建Z轴的旋转直接关键帧5Create Explicit Key Scale X创建X 轴的放缩直接关键帧Create Explicit Key Scale Y创建Y 轴的放缩直接关键帧Create Explicit Key Scale Z创建Z 轴的放缩直接关键帧Create Exposure Control创建曝光控制Create Extrude Surface创建拉伸表面；创建拉伸曲面Create Faces(Mesh)创建面数(网格) Create Fillet Curve创建倒圆角曲线Create Fillet Surface创建倒圆角表面；创建倒圆角曲面Create Fit Curve创建拟合曲线Create Key创建关键帧Create Lathe Surface创建旋转表面；创建旋转曲面Create Line创建线Create Mirror Curve创建镜像曲线Create Mirror Surface创建镜像表面；创建镜像曲面Create Mode创建方式Create Morph Key创建变形关键帧Create New Set创建新集合Create Normal Projected Curve创建法线投影曲线Create Offset Curve创建偏移曲线Create Offset Point创建偏移点Create Offset Surface创建偏移表面；创建偏移曲面Create Out of Range Keys创建范围外帧Create Parameters创建参数Create Point创建轴点Create Points创建点Create Point Curve创建点曲线Create Point Curve on Surface创建表面点曲线Create Point Surface创建点表面；创建点曲面Create Polygon创建多边形Create Polygons创建多边形Create Position Lock Key创建位置锁定时间Create Primitives创建几何体Create Rotation Lock Key创建旋转锁定时间Create Ruled Surface创建规则表面；创建规则曲面Create Shape创建截面Create Shape from Edges由边创建图形Create Surfaces创建曲面Create Surface-Curve Point创建表面-曲线点Create Surface Edge Curve创建表面边界曲线Create Surface Offset Curve创建表面偏移曲线Create Surface-Surface IntersectionCurve创建表面与表面的相交曲线Create Surf Point创建面点Create Transform Curve创建变形曲线Create Transform Surface创建变换表面；创建变换曲面Create U Iso Curve创建U Iso曲线Create U Loft Surface创建U放样表面；创建U放样曲面Create UV Loft Surface创建UV放样表面；创建UV放样曲面Create Vertex创建顶点Create Vertices创建顶点数Create Vector Projected Curve创建矢量投影曲线Create V Iso Curve创建V Iso曲线Create 1-Rail Sweep创建1-围栏Create 2-Rail Sweep创建2-围栏Creation Method创建方式Creation Time创建时间Crop切割区域；渲染指定的区域，图像大小为指定区域的大小Crop Selected切割选择；按选择对象的边界盒定义的区域渲染，图像大小为指定区域的大小Cross相交Cross Section交叉断面；截面；相交截面；截面参数Crossing横跨Crossing Selection横跨选择CrossSection交差截面；截面CrossSection Modifier交差截面修改器Crowd群体；群集Cube正方体；立方体Cube/Octa立方体/八面体Cubic立方Current当前；当前的Current Class ID Filter当前过滤类别Current Combinations当前组合Current Nodes当前节点Current Object当前物体Current Objects当前物体Current Targets当前目标Current Time当前时间Current Transform当前变换Curvature曲率Curve曲率；曲线；当前Curve Approximation曲线精度控制；曲线近似；曲线逼近Curve Common普通曲线Curve-Curve曲线对曲线Curve-Curve Intersection Point曲线对曲线求交点Curve Editor动画曲线编辑器；运动曲线编辑器Curve Editor(Open)运动曲线编辑器(打开)Curve Fit曲线适配Curve Point曲线点；曲线对点6Curve Properties曲线属性Curves曲线Curves Selected被选择的曲线Custom自定义Custom Attributes自定义属性；定制属性Custom Bounding Box自定义绑定物体；自定义边界盒Custom Colors定制颜色Custom Icons自定义图标Customize自定义Customize Toolbars自定义工具条Customize User Interface自定义用户界面Cut剪切Cut Edge剪切边Cut Faces剪切面数Cut Polygons剪切多边形CV Curve可控曲线CV Curve on Surface曲面上创建可控曲线；曲面上的可控曲线CV on Surf曲面CVCV Surf可控曲面CV Surface可控曲面CV Surface Object可控曲面物体CVs Selected被选择的可控节点Cycle循环Cycle Selection Method循环选择方法Cycle Subobject Level循环子物体级别Cycle Through Scale Modes通过放缩方式循环Cycle Vertices循环节点Cycles周期；圈；圈数Cyclic Growth循环增长；循环生长；周期增长CylGizmo柱体线框；柱体框Cylinder圆柱体Cylinder Emitter柱体发射器Cylinder Gizmo柱体线框Cylinder Object圆柱体Damper减振器；阻尼器Damper Dynamics Objects阻尼器动力学物体Dashpot SystemDay日Daylight日光Deactivate All Maps关闭全部贴图；取消激活所有视图Decay衰减Decimals位数Default缺省；缺省值；默认；默认值Default Lighting Toggle默认照明开关Default Projection Distance默认的投影距离Default Viewport QuadDefine定义Define Stroke定义笔触Deflector导向板Deflector Space Warp导向板空间变形Deflectors导向板Deform变形Deformation变形Deformations变形Deforming Mesh CollectionDeg度Degradation退化，降[减]低，减少，降格[级]，老[软]化degree角度；度数degrees度；角度Delaunay德劳内类型Delegate代表Delete删除Delete a Position Key on X在X轴删除位置关键帧Delete a Position Key on Y在Y轴删除位置关键帧Delete a Position Key on Z在Z轴删除位置关键帧Delete a Rotation Key on X在X轴删除旋转关键帧Delete a Rotation Key on Y在Y轴删除旋转关键帧Delete a Rotation Key on Z在Z轴删除旋转关键帧Delete a Scale Key on X在X轴删除放缩关键帧Delete a Scale Key on Y在Y轴删除放缩关键帧Delete a Scale Key on Z在Z轴删除放缩关键帧Delete All全部删除Delete All Position Keys删除全部位置关键帧Delete All Rotation Keys删除全部旋转关键帧Delete All Scale Keys删除全部放缩关键帧Delete Both删除行列Delete Button删除按钮Delete Col.删除列Delete Curve删除曲线Delete Key删除关键帧Delete Mesh Modifier删除网格修改器Delete Morph Target删除变形目标Delete Objects删除物体Delete Old删除旧材质；删除当前场景中的对象，合并新来的对象Delete Operand删除操作物体；删除操作对象Delete Original Loft Curves删除原放样曲线Delete Patch删除面片Delete Patch Edge删除面片边界Delete Patch Element删除面片元素Delete Patch Modifier删除面片修改器7Delete Patch Vertex删除面片节点Delete Row删除行Delete Schematic View删除图解视图Delete Segment删除线段Delete Shape删除图形Delete Spline删除曲线Delete Spline Modifier删除曲线修改器Delete Tab删除面板Delete Tag删除标记Delete the Pop-up NoteDelete Toolbar删除工具条Delete Track View删除轨迹视图Delete Vertex删除节点 Delete Zone 删除区域；删除色带Dens密度Density密度；强度；浓度DependenciesDependent Curves从属曲线Dependent Points从属点Dependent Surfaces从属曲面Dependents从属格线；关联Depth深度Depth of Field视野；景深Depth Segs深度片段数Derive From Layers来自层Derive From Materials来自材质Derive From Material Within Layer 来自层中的材质Derive Layers By导入层依据Derive Objects By导入物体方式；导入物体依据Derive Objects From导入物体依据Destination目的；显示出在当前场景中被选择对象的名字；目标位置Destination Time目标时间Destory CharacterDetach分离；从对象组中分离对象Detach Element分离元素Detach Segment分离线段Detach Spline分离曲线Details细节Deviation背离；偏差Dialog对话框Diameter直径Die After Collision碰撞后消亡Diffuse漫反射；漫反射光；表面色；过渡区Diffuse(reflective&translucent)过渡色(反射与半透明)Direction方向Direction Chaos方向混乱Direction of Travel/Mblur运动方向/运动模糊Direction Vector矢量方向；方向向量Directional方向；方向型Directional Light平行光Disable无效Disable Scene Redraw ToggleDisable View显示失效；视图无效Disable Viewport非活动视图Disable Viewport Toggle视图切换失效DisassembleDisassemble ObjectsDiscard New Operand Material丢弃新材质Discard Original Material丢弃原材质Disintigrate裂解Disp Approx位移近似Disp Approx Modifier位移近似修改器Displace置换；位移；位移编辑修改器Displace Modifier位移修改器Displace Space Warp位移转换空间变形Displaced Surface贴图置换表面；置换贴图表面；位移表面；位移曲面Display显示；当Display处于打开时，在绘图时会出现捕捉导线。
Parallax-tolerant Image Stitching
Parallax-tolerant Image Stitching Fan Zhang and Feng LiuDepartment of Computer SciencePortland State University{zhangfan,fliu}@AbstractParallax handling is a challenging task for image stitch-ing.This paper presents a local stitching method to handle parallax based on the observation that input images do not need to be perfectly aligned over the whole overlapping re-gion for stitching.Instead,they only need to be aligned in a way that there exists a local region where they can be seam-lessly blended together.We adopt a hybrid alignment model that combines homography and content-preserving warp-ing to provideflexibility for handling parallax and avoiding objectionable local distortion.We then develop an efficient randomized algorithm to search for a homography,which, combined with content-preserving warping,allows for op-timal stitching.We predict how well a homography enables plausible stitching byfinding a plausible seam and using the seam cost as the quality metric.We develop a seamfinding method that estimates a plausible seam from only roughly aligned images by considering both geometric alignment and image content.We then pre-align input images using the optimal homography and further use content-preserving warping to locally refine the alignment.Wefinally compose aligned images together using a standard seam-cutting al-gorithm and a multi-band blending algorithm.Our exper-iments show that our method can effectively stitch images with large parallax that are difficult for existing methods.1.IntroductionImage stitching is a well-studied topic[22].Itsfirst step is to align input images.Early methods estimate a2D trans-formation,typically a homography,between two images and use it to align them[23,3].Since a homography can-not account for parallax,these methods require that the in-put images should be taken from the same viewpoint or the scene should be roughly planar.Otherwise,no homogra-phy exists that can be used to align these images,resulting in artifacts like ghosting or broken image structures.While advanced image composition techniques,such as seam cut-ting[2,12]and blending[4,17],can relieve these artifacts, they cannot address significant misalignment.Recent image stitching methods use spatially-varying warping algorithms to align input images[13,27].While spatially-varying warping can better handle parallax than homography,it still cannot work well on images with large parallax.Figure1shows a challenging example with a sig-nificant amount of parallax in input images.Notice the hor-izontal spatial order of the car,the tree,and the chimney in the input images shown in Figure1(a).In the left input image,the chimney is in the middle of the car and the tree while in the right image,the tree is in the middle of the car and the chimney.For this example,one image actually needs to be folded over in order to align with the other.This is a fundamentally difficult task for the warping methods as they either cannot fold over an image or will bring in objec-tionable distortion,as shown in Figure1(c).In this paper,we present a parallax-tolerant image stitch-ing method.Our method is built upon an observation that aligning images perfectly over the whole overlapping area is not necessary for image stitching.Instead,we only need to align them in such a way that there exists a lo-cal region in the overlapping area where these images can be stitched together.We call this local stitching and de-velop an efficient method tofind such a local alignment that allows for optimal stitching.Our local stitching method adopts a hybrid alignment model that uses both homogra-phy and content-preserving warping.Homography can pre-serve global image structures but cannot handle parallax.In contrast,content-preserving warping can better handle par-allax than homography,but cannot preserve global image structures as well as homography.Moreover,local stitch-ing still prefers a well aligned,large local common region. However,when homography is used to align images with large parallax,the local region size and alignment quality are often two conflicting goals.We address this problem by using homography to only roughly align images and em-ploying content-preserving warping to refine the alignment.We develop an efficient randomized algorithm to search for a homography for inexact local alignmentfirst.Therein, we predict how well a homography enables local stitching byfinding a plausible seam from the roughly aligned im-(a)Inputimages(b)AutoStitch(c)APAP[27](d)Our resultFigure 1.Parallax problem in image stitching.All the results are partially cropped for the sake of layout.For input images with large parallax,homography-based methods,such as AutoStitch,cannot align input images and suffer from ghosting artifacts (b).Spatially-varying warping methods,such as [27],can align im-ages but introduce apparent visual distortion (c).Our method can produce an artifacts-free stitching result (d).ages and using the seam cost to score the homography.We develop a graph-cut based seam finding method that can es-timate a plausible seam from only roughly aligned images by considering both geometric alignment and image con-tent.Once we find the optimal homography,we use it to pre-align the input images and then use content-preserving warping to re fine the alignment.The main contribution of this paper is an ef ficient and robust stitching method that handles images with large par-allax well.The power of our method comes from local stitching,which,enhanced by content-preserving warping and seam cutting,explores both image content and geomet-ric alignment and finds an optimal local region to stitch im-ages together.As shown in our experiments,our method can stitch images with a signi ficant amount of parallax.2.Related WorkImage stitching has been well studied in the fields of computer vision and graphics.A good survey can be found in [22].This section only gives a brief overview and fo-cuses on parallax handling.Most existing image stitching methods estimate a 2D transformation,typically a homography,between two input images and use it to align them [23,3].These homography-based methods can work well only when the input images have little parallax as homography cannot account for par-allax.When input images have large parallax,artifacts like ghosting occur.Local warping guided by motion estima-tion can be used to reduce the ghosting artifacts [21].Image composition techniques,such as seam cutting [12,2,7]and blending [4,17],have also been employed to reduce the arti-facts.However,these methods alone still cannot handle sig-ni ficant parallax.We also use seam cutting and blending as the final steps in our method.The recent dual-homography warping method can stitch images with parallax,but it re-quires the scene content can be modeled by two planes [8].Multi-perspective panorama techniques can handle par-allax well [20,28,16,6,19,1,26,18].These techniques require 3D reconstruction and/or dense sampling of a scene.They are either time-consuming or cannot work well with only a sparse set of input images,as typically provided by users to make a panorama.The idea behind some of these multi-perspective panorama techniques inspired our work.That is,input images do not need to be perfectly aligned over the whole overlapping image region.As long as we can piece them together in a visually plausible way,a visu-ally pleasing panoramic image can be created.A relevant observation has also been made in a recent work that the best-fitting homography does not necessar-ily enable optimal image stitching [9].They estimate a set of homographies,each representing a planar structure,cre-ate multiple stitching results using these homographies,and find the one with the best stitching quality.This method can successfully handle parallax for some images and also inspired our work;however,it is slow as it needs to cre-ate and score multiple stitching results.More importantly,sometimes none of the homographies that represent some planar structures can enable visually plausible stitching.In contrast,our method evaluates the alignment quality without creating the stitching results and is more ef ficient.Also,our method integrates content-preserving warping and loosens the alignment requirement for homography estima-tion,thus providing more alignment candidates.Moreover, our method,considering both image content and geometric alignment in searching for a local alignment,can obtain an alignment that suits image stitching better.Recently,spatially-varying warping methods have been extended to image stitching.Lin et al.developed a smoothly varying affine stitching method to handle parallax[13]. Zaragoza et al.developed a technique to compute an as-projective-as-possible warping that aims to be globally pro-jective while allowing local non-projective deviations to ac-count for parallax[27].These methods have been shown to work well on images with parallax that are difficult for homography-based methods.However,they still cannot handle images with large parallax,as shown in Figure1. Our method also employs a variation of spatially-varying warping method,but only uses it to align input images overa local overlapping region.3.Parallax-tolerant Image StitchingOur method uses a common image stitching pipeline. Specifically,wefirst align input images,then use a seam cutting algorithm tofind a seam to piece aligned images to-gether[12],andfinally employ a multi-band blending algo-rithm to create thefinal stitching result[4].Our contribution is a novel image alignment method which can align images in such a way that allows for optimal image stitching.Our observation is that we do not need to perfectly align images over their whole overlapping area.In fact,for im-ages with large parallax,it is very difficult,if not impossi-ble,to align them perfectly.Our goal is to align images in a local region where we canfind a seam to piece them to-gether.We employ a randomized algorithm to search for a good alignment.Specifically,wefirst detect SIFT feature points and match them between two images[15].We then randomly select a seed feature point and group its neigh-boring feature points to estimate an alignment as our goal is to estimate an alignment that aligns images over a local region with a compact feature distribution.We evaluate the stitching quality of this alignment.If this alignment is de-termined good enough for stitching,we stop;otherwise we repeat the alignment estimation and quality evaluation.Be-low wefirst discuss some key components of this algorithm and then provide a detailed algorithm description.3.1.Alignment Model SelectionThefirst question is what alignment model to use.There are two popular options:global2D transformation,typi-cally homography,and spatially-varying warping,such as content-preserving warping[14,24].Most existing meth-ods use a global2D transformation to align two images.A global2D transformation has an important advantage in that it warps an image globally and avoids some objectionable local distortions.For example,homography can preserve lines and similarity transformation can preserve the object shape.But they are too rigid to handle parallax.For image stitching,while we argue that it is not necessary to align images exactly in their whole overlapping area,it is still preferable to align images well over an as large as possible common region.However,for images with large parallax, a2D transformation,even a homography,can often only align images over a small local region.In contrast,content-preserving warping is moreflexible and can better align im-ages,but it often introduces objectionable local distortion.Our solution is to combine these two alignment mod-els to align images well over a large common region with minimal distortion.Given a seed feature point,our method incrementally groups its neighboring feature points tofit a2D transformation(a homography by default).Here we use a slightly largefitness threshold in order to group as many feature points as possible although this makes the ho-mography unable tofit these feature correspondences ex-actly.Loosing thefitness of the homography can be com-pensated by applying content-preserving warping later on, as content-preserving warping is well suited to local warp-ing refinement without introducing noticeable distortion.3.2.Alignment Quality AssessmentA straightforward way to evaluate the stitching quality of the above mentioned hybrid alignment is tofirst warp an image using the homography and apply content-preserving warping.We can then compare the warped image and the reference image to examine how well these two images are aligned.This approach,however,cannot reliably predict whether a good seam can be found in the overlapping re-gion.Furthermore,this approach does not consider the ef-fect of image content on stitching.For stitching,salient im-age features,such as edges,should be well aligned while image regions like the sky do not necessarily need to be per-fectly aligned.Finally,this approach is slow as it needs to run content-preserving warping whenever we evaluate the alignment quality inside the randomized algorithm.We address the above problems as follows.First,we examine the alignment quality based on the image edges in-stead of the raw image directly.Second,we only evaluate how the homography supports stitching.This simplifica-tion can be justified by the fact that content-aware warp-ing is very effective if only minor adjustment to the global warping is required.But it also brings in a challenge:the homography in our method is designed to be loose and does not align two images exactly.Then we need to predict how well the alignment enables seamlessly stitching from only roughly aligned images.We address this challenge byfind-ing a plausible seam from the roughly aligned images and using the seam cost to score the alignment.Wefirst down-sample the input images to both improve speed and tolerate the small misalignment.We then com-(a)Inputimages(b)Optimal local homographyalignment(c)Content-preservingwarping(d)Stitching resultFigure2.Stitching pipeline.Please zoom in thisfigure to better examine the alignment results at(b)and(c).Given input images with large parallax(a),our methodfirst estimates an optimal homography that roughly aligns images locally(b)and is predicted to allow for optimal stitching as described in Section3.2.In(b)and(c),we only blend aligned images by intensity averaging to illustrate alignment.The red and green points are the SIFT feature points in the warped image and the reference image,respectively.When two feature points are aligned,they appear olive green.Only a subset of feature points,indicated by blue circles,are selected tofit a homography loosely.Our method then locally refines alignment using content-preserving warping(c),andfinally employs seam-cutting and multi-band blending to create thefinal stitching result(d).pute the edge maps for the input images using the Cannyedge detection method[5].The edge maps are low-passfil-tered to tolerate the small misalignment.We compute thedifference between the warped edge map and the referenceimage’s edge map and obtain the difference map E d.Aplausible seam should avoid passing pixels with large valuesin the difference map in order to obtain a seamless stitchingresult.We extend the graph-cut seamfinding method[12]tofind a plausible seam.Briefly,we consider each pixel inthe overlapping region as a graph node.We define the edgecost between two neighboring nodes s and t as follows,e(s,t)=f c(s)|E d(s)|+f c(t)|E d(t)|(1)where we use an alignment confidence function f c(s)toweight the edge cost.f c(s)is computed to further accountfor the fact that the homography can only align two im-ages roughly and content-preserving warping will be usedto refine the alignment.Specifically,if a local region hasa SIFT feature point,the alignment there can very likelybe improved by content-preserving warping and thus themisalignment from only using the homography should bedeemphasized.We compute f c(s)to deemphasize the mis-alignment according to the SIFT distribution as follows,f c(s)=1P ig( P s−P i )+δ(2)where P i is the position of a SIFT feature point and P s is the position of pixel s.g is a Gaussian function and is used to propagate the effect of a SIFT feature to its neighborhood.δis a small constant with a default value0.01.Based on the edge cost defined in Equation1,the seam finding problem can be formulated and solved as a graph-cut problem[12].Once we obtain this seam,we use the cost associated with this seam to score the alignment quality. 3.2.1Homography ScreeningWhile some homographies can allow for seamless stitch-ing,they sometimes severely distort the images and lead to visually unpleasant stitching results.We detect such homo-graphies and discard them before evaluating their alignment quality.We measure the perspective distortion from apply-ing a homography H to an image I by computing how H deviates from its best-fitting similarity transformation.De-note C i as one of the four corner points of the input imageI and¯C i is the corresponding point transformed by H.We find the best-fitting similarity transformationˆH s as follows,ˆHs=arg minH sC iH s C i−¯C i 2,where H s=a−b cb a d(3)Once we obtainˆH s,we sum up the distances between the corner points transformed by H andˆH s to measure the per-spective distortion.If the sum of the distances normalized by the image size is larger than a threshold(with default value0.01),we discard that homography.3.3.Alignment Algorithm SummaryWe now describe our randomized algorithm to estimatea good alignment for stitching.1.Detect and match SIFT features between input im-ages[15]and estimate edge maps for input images[5]. 2.Randomly select a seed feature point and group its spa-tially nearest neighbors one by one until the selectedfeature set cannot befitted by a homography with apre-defined threshold.We maintain a penalty value foreach feature point to identify the times that it has beenselected during the iteration process.When a featurepoint is selected,we increase its penalty value by one.In each iteration,to be selected as a valid seed,a fea-ture point should not have been selected as a seed be-fore and its penalty score is below the average penaltyvalue of all the feature points.3.Evaluate the alignment quality of the best-fitting ho-mography from Step2using the algorithm described inSection3.2.If the homography meets the pre-definedquality threshold,go to Step4.Otherwise,if the av-erage penalty value is low,go to Step2;otherwise se-lect the best homography estimated during the iterationprocess and go to Step4.4.Employ the optimal homography to pre-align imagesand use content-preserving warping guided by the setof selected feature points to refine the alignment,asdescribed in Section3.3.1.Figure2shows the pipeline of our method.Given inputimages(a),our methodfirstfinds an optimal local homog-raphy and a subset of feature points that are looselyfit bythis homography as shown in(b).We illustrate the selected feature pairs using blue circles.Notice that the homographydoes not align these features exactly.We then use contentpreserving warping to refine the alignment.As shown in(c), the selected feature pairs are now well aligned.Our methodfinally composes the aligned images together(d).3.3.1Content-preserving warpingVarious content-preserving warping methods have beenused in applications,such as video stabilization[14]and image and video retargeting[25,24].While content-preserving warping alone cannot always be used to alignimages over their whole overlapping area,it is well suited for small local adjustment.Therefore,we use it to furtheralign the pre-warping result from the optimal homographyto the reference image as shown in Figure2(b)and(c). We use I,¯I,andˆI to denote the input image,the pre-warping result,and thefinal warping result,respectively. We divide the input image I into an m×n uniform gridmesh.The vertices in I,¯I,andˆI are denoted using V i,¯V i, andˆV i.We then formulate the image warping problem as a mesh warping problem,where the unknowns areˆV i.¯V i is known from pre-warping.This mesh warping problem is defined as an optimization problem that aims to align¯I to the reference image while avoiding noticeable distortions. We now describe the energy terms in detail below.Local alignment term.The feature points in image Iand¯I should be moved to match their corresponding posi-tions in the reference image so that they can be well aligned. Since a feature point P j is not usually coincident with any mesh vertex,wefind the mesh cell that contains P j.We then represent¯P j,the corresponding point of P j in¯I,using a linear combination of the four cell vertices of the corre-sponding cell in image¯I.The linear combination coeffi-cients are computed using the inverse bilinear interpolation method[10].These coefficients are used to combine the vertices in the output imageˆI to computeˆP j.We can then define the alignment term as follows.E p=nj=1αj,kˆV j,k−˜P j 2,(4)where n is the size of the selected feature set from the align-ment optimization step(Section3.3),αj,k is the bilinearcombination coefficient,andˆV j,k is a vertex of the meshcell that containsˆP j,and˜P j is the corresponding featurepoint in the reference image.Global alignment term.The alignment term above onlydirectly constrains warping of the overlapping image regionwith selected feature points.For other regions,content-preserving warping often distorts them.As the pre-warpingresult¯I has already provided a good approximation,ourmethod encourages the regions without feature points to beclose to the pre-warping result as much as possible.Wetherefore define the following global alignment term,E g=iτi ˆV i−¯V i 2,(5)whereˆV i and¯V i are the corresponding vertex in the content-preserving warping result and in the pre-warping result.τiis a binary value.We set it1if there is a feature point inthe neighborhood of V i;otherwise it is0.This use ofτiprovidesflexibility for local alignment.Smoothness term.To further minimize the local distor-tion during warping,we encourage each mesh cell in thepre-warping result to undergo a similarity transformation.We use the quadratic energy term from[11]to encode thesimilarity transformation constraint.Specifically,considera triangle ¯V1¯V2¯V3.Its vertex¯V1can be represented by theother two vertices as follows,¯V1=¯V2+u(¯V3−¯V2)+vR(¯V3−¯V2),R=01−10,(6)where u and v are the coordinates of¯V1in the local coordi-nate system defined by¯V2and¯V3.If this triangle undergoesa similarity transformation,its coordinates in the local coor-dinate system will not be changed.Therefore,the similaritytransformation term can be defined as follows,E s(ˆV i)=w s ˆV1−(ˆV2+u(ˆV3−ˆV2)+vR(ˆV3−ˆV2)) 2,(7)where u and v are computed from Equation6.We sumE s(ˆV i)over all the vertices and obtain the full smoothnessenergy term E s.Here w s measures the saliency value ofthe triangle ¯V1¯V2¯V3using the same method as[14].Weuse this saliency weight to distribute more distortion to lesssalient regions than those salient ones.Optimization.We combine the above three energyterms into the following energy minimization problem,E=E p+αE g+βE s,(8)whereαandβare the weight of each term with default val-ues0.01and0.001,respectively.The above minimizationproblem is quadratic and is solved using a standard sparselinear solver.Once we obtain the output mesh,we use tex-ture mapping to render thefinal result.4.ExperimentsWe experimented with our method on a range of challenging images with large parallax.We also com-pared our method to the state-of-the-art methods,includingPhotoshop,AutoStitch,as-projective-as-possible stitching (APAP)[27],and our implementation of seam-driven stitch-ing (SEAM)[9].For APAP,we used the code shared by the authors.Since that code only aligns images,we applied the same seam-cutting and multi-band blending algorithm used in our method to the APAP alignment results to produce the final stitching results.This paper only shows some repre-sentative stitching results that are partially cropped for the sake of layout.Please refer to the project website for more results that are not cropped and more intermediate results 1.Figure 3(a)shows two input images with a signi ficant amount of parallax.Photoshop failed to produce any re-sult.AutoStitch could not align two images well using a global 2D transformation,therefore the stitching result suf-fers from ghosting,as indicated by the red circle in Fig-ure 3(b).The traf fic light is duplicated in the final result.The SEAM method did not find a local plane represented by a homography that allows for seamless stitching,and duplicated the traf fic light too as shown in Figure 3(c).The APAP method creates a reasonable stitching result as shown in Figure 3(d);however,as APAP tries to align two images over the whole overlapping region,it distorts the salient im-age structure,such as the pillar indicated by the red rect-angle.Our method can handle this challenging example by aligning the input images locally in a way that allows for optimal stitching,as shown in Figure 3(e).We also show the stitching seam in red.Figure 4(a)shows another challenging example.The two input images have a large amount of parallax,and there is no global transformation that can align them well over the whole overlapping region.As shown in Figure 4(b),the AutoStitch result suffers from signi ficant ghosting artifacts.While blending can relieve misalignment,it causes severe blurring artifacts as indicated by the red circle.Both Pho-toshop and SEAM duplicated the red structure,as shown in Figure 4(c)and (d).APAP bends the straight line as shown in Figure 4(e).Our result in (f)is free from these artifacts.4.1.DiscussionOur method only needs to align input images locally and fit a homography loosely,as described in Section 3.2.Therefore our method can sometimes use a more restric-tive global transformation than a homography to remove the perspective distortion from homography.Figure 5(a)shows a stitching result from our method using homography for initial alignment,which suffers from noticeable perspective distortion.Once we replace homography with similarity1/project/stitch(a)Input images(b)AutoStitch(c)SEAM [9](d)APAP [27](e)Our result with seamFigure parisons among various stitching methods.transformation for initial alignment,the stitching result suf-fers from less distortion,as shown in Figure 5(b).We also tested how the homographies selected by our method differ from the best-fitting ones by computing the distances between the transformed image corner positions with our homographies and the best-fitting ones.Over 75%of the examples shared in our project website has the av-(a)Input images (b)AutoStitch(c)Photoshop (d)SEAM[9](e)APAP [27](f)Our resultFigure parisons among various stitching methods.erage corner position distance larger than 36pixels (given an image with width 1000pixels).The median distance is around 60pixels.This con firms that our method uses dif-ferent homographies than the best-fitting ones.Our method works well on a range of examples with large parallax as well as all the examples reported in the recent APAP paper [27].Meanwhile,we also found some failure cases as shown in the project website.One was that input images have very large parallax and are full of salient structures.For stitching,images must be aligned so that there at least exists a local common region where a good seam can be found.In images with large parallax,there is often no such a local region that can be aligned.Our method explores the fact that non-salient areas often need not be well aligned and considers this in searching for a good local region alignment.But if an image has large parallax and is full of salient structures,our method sometimes cannot work as no non-salient region exists.Our method adopts a common image stitching pipeline.Its major novelty is in its step to align images such that op-timal stitching can be achieved.This step,including op-timal local homography estimation and content-preserving warping,typically takes from 20to 40seconds on a desk-top machine with Intel i7CPU and 8GB memory to align two images with width 1000pixels.All the other steps are shared by off-the-shelf image stitching methods.5.ConclusionThis paper presented a parallax-tolerant image stitching method.We observed that images with signi ficant parallax often cannot be aligned well over the whole overlapping re-gion without suffering artifacts like folding-over and these images actually do not need to be aligned perfectly over the whole overlapping region for image stitching.We then de-veloped a method that aligns input images locally in such a way that allows for optimal stitching.We designed an ef ficient algorithm to estimate how an alignment result en-ables seamless stitching without actual stitching.Our ex-periments on challenging stitching tasks showed the effec-tiveness of our method.。
计算机图形学latest03
θ φ
x' = r cos(θ + φ ) = r cos φ cosθ − r sin φ sin θ y ' = r sin(θ + φ ) = r cos φ sin θ + r sin φ cosθ
(x,y) r
x
Q x = r cosφ , y = r sin φ
∴ x ' = x cos θ − y sin θ , y ' = x sin θ + y cos θ
0 Sy 0
0 x 0 y Sz z
Scaling can make an object bigger or smaller, as shown in the left figure, which illustrates both uniform scaling in all directions and scaling in a single direction.
a 11 x y = a 21 a 31 z a 12 a 22 a 32 a 13 x x a 23 y = A y z a 33 z
x x x y = A −1 y = A T y z z z
and maps the point into the corresponding point within another coordinate system.
4
Transformation Pipeline
Modeling Coordinates Modeling Transformation World Coordinates Viewing Transformation Viewing Coordinates
极角 方位角 英语
极角方位角英语Polar Angle and Azimuth in EnglishThe concept of polar angle and azimuth are fundamental to understanding the spatial positioning of an object in athree-dimensional coordinate system.The polar angle,also known as the zenith angle or inclination angle,represents the angle between the positive z-axis and the vector that points to the object.It is measured in degrees or radians and ranges from0°to180°.On the other hand,the azimuth angle,also known as the bearing or horizontal angle,represents the angle between the positive x-axis and the projection of the vector onto the xy-plane.It is measured in the xy-plane from the positive x-axis and ranges from0°to360°.In practical applications,understanding the polar angle and azimuth of an object is crucial for various fields such as astronomy,navigation,and robotics.By knowing these angles,one can determine the exact orientation of an object relative to a reference point or coordinate system.In conclusion,the concepts of polar angle and azimuth play a key role in defining the spatial position of an object in a three-dimensional space.By understanding and applying these concepts,one can accurately describe and analyze the orientation of objects in a variety of fields.。
Lathe
In some latitudes, a belt drive system is used to transmit power from the motor to the spindle, promoting a cost effective and related solution
Characteristics
The main characteristics of a Lathe include high precision, high efficiency, wide applicability, and easy operation Modern lathes also have features such as automatic tool change, automatic feeding and speed control, and programmable control systems
Regular maintenance
Regular maintenance, including cleaning and lubrication, helps to maintain the accuracy of the over time
Calibration
Periodic calibration of the Lathe's components helps to ensure that they are operating within specified tolerances
Application fields and market demand
Application fields Lates are widely used in the manufacturing industry for machining variant types of workpieces such as shares, disks, and complex shapes They are also used in the automotive industry for machining engine blocks, cylinder heads, and other components In addition, these are used in the aerospace industry for machining precision parts such as turbine blades and landing gear components
拉深件英文原文及翻译
Study of the Parameters of Deep Drawing Process Based on theSimulation of DynaformKeywords: Deep Drawing, Blank Holder Force; Deep Drawing Speed, Forming Speed, Friction Coefficient.Abstract:A simulation of deep drawing process on the sheet metal was done by using Dynaform; the influence of blank holder force, deep drawing speed and friction coefficient on the forming speed of sheet metal in the deep drawing process were got. The forming speed of sheet metal determines the quality of deep drawing, in the deep drawing process the blank holder force and the deep drawing speed are controllable parameters, the friction coefficient can be intervened and controlled, and it's a manifestation of the interaction of all parameters, the main factors which influence the friction coefficient just have blank holder force, deep drawing speed and lubrication except the material. The conclusion of this study provides the basic data for the analysis of the lubrication of mould, the study of lubricant arid the prediction of the service life of deep drawing die.Introduction:The sheet metal deep drawing is a very important method of metal plastic forming, which widely used in automotive, aviation, aerospace, etc. With the rapid development of modern industry, the research of deep drawing focuses on theories and simulation analysis, the main content of research is the influence of the deep drawing parameters on the deep drawing quality. Theoretical analysis mainly includes the derivation of analytic solutions under certain conditions. Document simplifies the format of axial symmetry problem's radial stress. Document utilizes grey system theory, takes the relationship between target sequences as the target function, transforms mufti-objective optimization into single-objective optimization. The robust design model of sheet forming is built under multiple targets, such as not resulting in crack, not resulting in wrinkle, and enough deforming, and it also need be met that the variation of responses should be Less.’ Document sets up a stress analytical formula for the dangerous section of the cylindrical parts' deep drawing. Taking the blank holder force which appears when the dispersibility deep drawing loses stability to be the critical blank holder force, it establishes an analytical formula of relative critical blank holder force. Document proposes the concept of the critical relative thickness of key ring, derives an engineering analytical formula which can determine the critical relative thickness. Document gives the fracture critical blank holder force theory which can be applied to square box through the research of fracture critical blank holder force curve and the experimental verification on the corresponding results. Document analyses the essence of the hydrodynamic deep drawing form with hybrid blank holder in the condition that the constant blank holder force set under fixed gap, and studies the influence of the changes of the blank holder gap, the blank holding force and the radial pressure on the forming process by using the finite element method. Document proposes a new hydraulic-pressure augmented deep drawing with fluid assisted blank holder. The radial pressure augmented by two pressure pistons is loaded onto the blank rim and the top surface of the blank flange, so the process of hydraulic-pressure augmented deep drawing with fluid assisted blank holder is realized. In order to reduce the frictional resistance on the interface between the blank flange and the draw die, the augmented liquid is also filled in the holes made on the certain part of the draw die surface. Document compares the drawing load variation and maximum drawing load in forming pure copper conical-cylindrical cups with the thickness of 2.5 mm by hydroforming and conventional multistage deep drawing processes by experiment. The results demonstrate that drawing loadvariation is more uniform in the forming of conical parts饰hydroforming deep drawing process. The. maximum drawing load for drawing copper blank occurs at a higher amount in hydroforming process. Document identifies the optimal position far the insertion of pressurized lubricant, predicts the potential of the hydrostatic pressure lubrication, and approves the results of the simulations with experimental tests after the numerical analysis of the appearing loads of deep drawing processes. Document verifies experimentally the warm deep drawing process assisted with hydraulic counter pressure as a suitable alternative to conventional deep drawing as a means for producing defect-free sheet metal parts. The hydraulic counter pressure helps in.reduction of wrinkles and enhancement of formability. The research of Documents focuses on the deep drawing technology and simulation of different kinds of deep drawing parts, carries out the simulation analysis and experimental treatments of cylindrical parts, rectangular case's and conical body based on Dynaform, gets the influence of blank holder force, blank shape and material performance parameters on the forming limit; Technological parameters of four main factors for double pass deep drawing of the cylindrical parts have been numerical simulated through- Dynaform, which results showed that the influencing regularities of varied factors upon drawing coefficient of the metal board are different for different deep drawing; The deep drawing performance of conical parts is influenced by many factors, such as the performance parameters of material, blank holder force, the friction between blank and die, die parameters,and the shape of parts; The deep drawing forming process of the flanged cylindrical parts is simulated, and the parts' distribution law of strain and reducing ratio of wall thickness which occur in deep drawing process is got.The purpose of this study is to provide the law of forming speed for the lubrication analysis, the main parameter which affects the lubrication effect is the relative velocity of each point's surface, and that would determines if the oil film can be formed. The forming speed is related to blank holder force deep drawing speed, and the friction characteristics coefficient of sheet metal; so this paper mainly simulates the influence of controllable parameters and friction characteristics on the forming speed. The lubrication conditions can be analyzed, and the die life can be predicted based on this, the excellent deep drawing quality and high efficiency can be got through adjusting friction condition and choosing lubricant at the same time.Establish the Siaalaalation ModelDetermine the sheet metal: the thickness is 1 mm, the material is St I4F cold rolled sheet, the diameter of blank is 106 mm. The shape after deep drawing is shown in figure 1.The figure 1 is established by using SolidWorks, the model is imported into Dynaform, then mesh the sheet metal, die and blank holder and check. The mesh generation affects the precision of simulation and the computing time, it should be meshed thick in the large deformation area, the area which has no deformation or small deformation can bethin. Insure the accuracy of the sheet metal simulation results, the mesh of sheet should be thick, the mesh of-die and blank holder can be thin. The sheet metal, blank holder and concave-convex die are adjusted to the positions shown in figure 2.The material parameters of deep drawing parts: density is 7.85x103kg/m3, Young modules is 2.07x105Pa, Poisson's ratio is 0.28 yield strength is 350MPa, in-plane anisotropy coefficient is Roo=1.73,R45=1.35, R9o=2.18.Analysis of the Deep Drawing Process ParametersIn order to analyze and get effective process parameters, the main factor which influences the quality of deep drawing is velocity parameters except the design of die itself. The deep drawing speed affects the deformation rate of drawing parts, the deformation rate determines the relative motion state between drawing parts and die, the relative motion affects the lubrication of deep drawing process, the lubrication is directly related with the surface quality after deep drawing. Of course, the choice of speed can directly affect whether can farming, so it is the most important parameter of deep draining process. Choose various speeds to simulate in order to analyze, compare, and get good practical process parameters.The die's movement of deep drawing can be divided into two stages; the first is pressure stage, in the stage the die moves downward to the blank holder and forms an effective pressing force with the sheet metal, forms the so-called blank holder force; The second is stamping stage, the die continues to move in a predetermined speed until the deep drawing process is completed. In the pressure stage, the speed of movement is 2m/s, do the simulative calculation by choosing 3.Sm/s, 4.0 m/s, 4.5 m/s, 5.0 m/s, 5.5 m/s and 6.0 m/s as the speed of deep drawing stage, 60KN, 80KN, 100KN, 120KN, 140KN and 160KN as the blank holder force, 0.05, 0.075, 0.1, 0.125, 0.15 and 0.175 as the friction coefficient respectively.Deep Drawang speed and F.rrning speedIn the deep drawing process of sheet metal, the change law of the metallic plastic deformation resistance isIn the equation: c—the deformation resistance when =1—the strain ratem—the strain rate coefficients, indicates the uniform deformation ability of sheet metal when the deep drawing speed changes, the bigger the m，the better the uniform deformation ability:In the deep drawing process, the increase of deep drawing speed will cause the decrease of the plasticity of sheet metal. The node numbers and positions which researched in the analysis are shown in the figure 3. The three positions whose note numbers are 4694, 5963 and 5977 of blank holder area were selected as the research object, represents the motion features of the blank holder area basically.The simulation result of forming speed is shown in the figure 4 and figure 5. The abscissas of the figures denote time, the unit is s; the ordinates denote the forming speed of notes, the unit is mm/s. In order to save the length of this text and be able to explain the change law, only take the speed of 3.5 m/s and b m/s to analyze. The figure 4 is the simulation result of each node in the speed of 3.5 m/s, and the figure 5 is the simulation result of each the same nodes in the speed of 6 m/s.From the above simulation figures; it is known that the variation trend of forming speed of the each node which belongs to the blank holder area is similar, the forming speed of the outer edge nodes is smaller than that of the inner edge notes. With the increase of the deep drawing speed, the forming speed increases: In the forming process, the size in thickness direction is smaller than that in radial direction, so the forming speed in thickness direction is smaller than that in radial direction and tangential direction. The forming speed represents radial forming speed and tangential forming speed in the figures. The deep drawing process of sheet metal can be realized clearly through the figures, the blank holder compress the sheet metal in the anterior few time steps, compress to the female die direction under the effect of the blank holder area, the punch keeps still, the oil film thickness of the blank holder area pressure decreases constantly at this .time until it can undertake the blank holder force, the sheet metal does not have the amoeboid locomotion in the radial direction but have the flow in the thickness direction, so the forming speed is small. After the completion of the pressure stage; the sheet metal enters the stable deep drawing forming stage and flows to the female die cavity with the driving of the punch, the sheet metal occurs bending deformation first, enters the female die cavity and then occurs straightening because of the effect of female die. In the process the flow speed of sheet metal increases rapidly and tends to be stable until the end of the forming process.The figure 6 describes the relation between the forming speed and the deep drawing speed of different nodes, the data in the figure is the value at the same time. It can be seen from the figure 6 that the forming speed in different positions is different at the same time, the blank holder area which is more close to the edge need to guarantee that the outer edge of blank holder area keeps fixedly due to the setting of blank holder force, so forms the simulation result shown in the figure. With the increase of deep drawing speed, the forming speed increases, it is easy to form the oil film lubrication in the view of tribology, and it is beneficial to improve the surface processing quality. The increase of the forming speed is easy to cause the tensile fracture. So the deep drawing speed must be controlled.The Influence of Blank Molder Force on the Forming SpeedThe blank holder force is an important factor that affects the sheet metal forming quality, in the forming process the main consideration of parameters includes the material parameters of sheet metal, shape parameters, the blank holder force; deep drawing speed, etc. The excessive blank holder force would cause the sheet metal cracking, namely, the sheet metal deformable area reduces and the sheet metal forming speed increases in the process. The undersized blank holder force would make the flange metal wrinkle.In order to study the influence of blank holder force on the forming speed, do the simulation of the notes of blank holder area by using 60KN, 80KN, I OOKN, 120KN, 140KN and 160KN as the blank holder force, 5m/s as the deep drawing speed. In order to save the length of this text, choose the two working conditions shown in the figure 7 and the figure 8.It can be seen from the figures that in the pressure stage the blank holder continues to move along the thickness direction after closing to the sheet metal; so the forming speed of each node is smaller and the influence of blank holder force is not obvious. In the deep drawing stage, the sheet metal flows to the female die with the effect of the punch, and occurs friction inevitable with the effect of blank holder because of the radial velocity of metal, thus it would restrain the metal flow, and the greater the blank holder force; the bigger friction of sheet metal receives, shown in the figures as: The metal flow trend of each forming area nearly keeps consistent, and slows down with the increase of bland holder force. When the blank holder force increases to a certain degree, the influence of it on the metal flow of blank border area reduces gradually, it shows that this time the blank holder force has affected the sheet metal forming, and it is easy to cause the cracking of sheet metal vertical wall.The curve of the relation at the same time between blank holder force and farming speed is shown in figure 9, it can be seen from the figure that the blank holder force has a stopping effect on the forming speed. The blank holder force can control the flow velocity of metal and influence the uniformity of forming. In order to avoid the wrinkling that caused by the uneven deformation of the metal of blank holderarea, the blank holder force should be added suitably, but the friction force of interface would be increased at the same time, the tensile stress of dangerous section would be increased when the blank holder force is too large, and that would cause the cracking or serious thin. So a reasonable selection of blank holder force is important to ensure the deep drawing forming quality of parts. In order to make the metal flow uniformly, the blank holder force should be tried to take the little value under the premise that the sheet metal does not wrinkle.The Influence of Friction Coefficient on the Forming SpeedIn the process of forming, the relative motion exists between sheet metal and die, the sheet metal produces plastic deformation, and the friction between the sheet metal and die or blank holder is produced. During the process the lubrication conditions a friction hinders the plastic flow, it is easy to cause the rupture of workpiece due to the increase of the needed external force and the tensile stress of workpiece. The friction between blank and die is to scratch the workpiece, lower the processing quality and the service life of die.In order to research the influence of the friction characteristics on the forming speed in the forming process, ao the slmmanon analysis by choosing0.05,0.75,0.1, 0.125, 0.15, 0.2 and 0.25 as the friction coefficient, and the nipture of sheet metal occurs when the friction coefficient reaches 0.25, so just choose the front six conditions to research. Take two kinds friction coefficient such as 0.05 and 0.15, the law curves of that are shown in the figure 10-11.It is can be seen from the figure 10-11 that the integral deformation with the increase of friction coefficient, just a slight change in value. It is 12 that the forming speed of each node of blank holder area decreases law remains unchangeably can be seen from the figure obviously with the increase of friction coefficient. The friction coefficient has great influence on the flow speed of sheet metal, so the flow ofmetal in the forming process can be controlled by using the friction between the sheet metal and die in practice; and the beneficial forming speed can be got.In the forming process the friction condition factors. In the blank holder area the friction is very complex and there are so many influencing speed, lubricant, environment condition coefficient is the integrated performance condition is related to the blank holder force, forming and the metal construction of sheet metal. The friction of friction condition, it depends on conditions the more and the lubrication conditions depend on forming speed. The larger beneficial the oil film to be formed, the smaller the friction effective lubrication the forming speed, coefficient and the impacts of blank-holder force similar to the effect coefficient. So the friction of the greater the friction coefficient, forming the lower speed. So the influence of friction coefficient is similar to the influence of blank holder force, the larger the friction coefficient, the lower the forming speed. Meanwhile, the friction status of blank holder area is directly related to the blank holder force, the friction between the sheet metal and the blank holder or female die increases inevitable with the increase of blank holder force.It is easy to cause cracking due to the overlarge friction between sheet metal and female die or blank holder, the decrease of the friction between the die of blank holder area and sheet metal is benefit for that the metal flows to female die, and restraining the attenuation of sheet metal vertical wall. So it is important to have a good grasp of the friction characteristics and take appropriate lubrication measures to improve the quality of the workpiece in the forming process.ConclusionsDid the simulation analysis of cylindrical parts by using Dynaform; and got the influence law of the controllable parameters on the forming result in the deep drawing process, the parameter selection of deep drawing decides the sheet metal forming quality, whether the qualified parts can be successfully produced depends on the design of parameters in engineering design. The purpose of this study is to get the influence of each control parameter on the forming, and then provide reference for the parameter design. The study got the influence of blank holder force, deep drawing speed and friction coefficient on the forming speed. The three parameters are controllable factors except the material, the design of the combination of hat canget a good effect, prolong the service life of die effectively and improve the quality of the surface. The parameters can be analyzed by using the tribological problems applied in the deep drawing process, at the same time, the law of forming speed can be got, simplify to be a speed equation, do the lubrication analysis successively by using the data simulation.。
Appearance-Mimicking Surfaces
Christian Sch¨ uller Daniele Panozzo ETH Zurich Olga Sorkine-Hornung
Figure 1: A collection of appearance-mimicking surfaces generated with our algorithm.
image, which has a strong relation with the depth buffer used in the standard graphics pipeline. Most works have proposed to create bas-reliefs from given digital 3D scenes by either directly compressing the depth buffer of the scene’s rendering or by working in the gradient domain, where the final model is obtained by solving a Poisson equation. In this work, we define appearance-mimicking surfaces (AMS) that generalize bas-reliefs, lifting their restriction to a height field. Our generalization makes the reliefs usable at a wider range of viewing angles, while still guaranteeing self-intersection free results, which is mandatory for subsequent fabrication. Specifically, we develop a mathematical framework to compute surfaces whose normals optimally approximate the normals of a given 3D shape or scene, while strictly obeying given depth- or volume-confinement constraints. Direct fitting of normals and spatial constraints is in general a challenging, nonlinear problem, which led previous works to employ heuristics that circumvent difficult numerical optimizations. Unfortunately, giving up the constrained optimization of normals means forfeiting bounds on geometry and appearance distortion in the resulting relief. Instead, we propose a novel view-dependent surface representation which allows us to cast the optimization as a quadratic program. The resulting problem formulation is convex, and we are guaranteed to find the optimal solution under feasible constraints. Differently from previous works, our method does not rely on rasterization of the input geometry and the depth buffer. AMS are generated by deforming the input mesh without modifying its connectivity, thereby increasing the algorithm’s efficiency, details preservation and allowing to easily transfer surface attributes. As a positive side effect of our representation, we can exactly satisfy per-vertex depth constraints and we can “project” the target shapes on disconnected and arbitrarily shaped surfaces, as shown in Figure 3. Our algorithm is controllable and robust, enabling to design complicated appearance-mimicking surfaces with minimal user effort. We test our method in a variety of applications, such as the design of optical illusions in architectural settings and the creation of carving patterns on complex geometries. To verify the realism of our model and lighting assumptions, we validate our results via 3D printing.
The thickening an operation for animation
Thickening:an operation for animationBy Sylvain Brandel,Dominique Bechmann*,Yves Bertrand..........................................................................................In this paper,we present a modelling operation dedicated to animation.This operation,called thickening,is a generalization of the extrusion operation.It allows to build from a graph 3-D objects with circular section and 4-D objects with spherical or toric section.Moreover,the date and the number of mergings and splittings in the associated animation are perfectly controlled,and the trajectory of each of the animated objects is completely established.Several types of thickening are presented in this paper,to build a surface or a volume,starting from a graph drawn in a surface or in a volume.Copyright #2000John Wiley &Sons,Ltd.Received:May 1999;Revised:19April 2000KEY WORDS :animation;topologically based modelling;4-D objects;thickeningIntroductionIn the ®eld of topologically based modelling for animation,we particularly study the topological modi®cations occurring during animations.These modi®cations are for instance merging and splitting,appearing and disappearing,or hole grooving and ®lling.Figure 1represents one of these animations,in which a topological sphere is transformed into a torus,next into a two-hole torus and ®nally into two toruses.Splitting an object into two other symmetrical objects is easy,but splitting an object into three or more objects is complex,as is the control of the behaviour of all these objects.A ®rst goal is to propose animations with multiple modi®cations.Each of these modi®cations allows to transform any number of objects in any number of other objects,and to control the trajectory of each object.Moreover,it is necessary to control the exact date of these modi®cations,which is always dif®cult to predict.This work provides an ef®cient and powerful frame-work for modelling 3-D and 4-D objects in order to build 2-D and 3-D animations.It allows to precisely control the topology,by splitting merging the object or creating holes.In the animation context,the method gives or full control of the number of splittings,their time of creation and their trajectory.The originality ofthe version of thickening presented in this paper is to use the space-time objects principle.The next section presents the principle of thickening and describes objectives and limitations of the method.Basic concepts and previous work are then discussed.The 3-D thickening (building of a surface)and 4-D thickening (building of a volume)are described for each type of graph support.Two extensions of thickening are presented,®rst a technique allowing to smooth the built objects and next a method to build objects with different radii between the thickened edges.The ®nd section concludes this work.Presentation of the ThickeningOperationThe thickening operation presented in this paper achieves the goal of generalizing the extrusion opera-tion to 2-D and 3-D graphs.This operation builds a surface (3-D thickening)or a volume (4-D thickening)`around'a graph.A graph is a set (not necessarily connected)of picked edges either on a surface (a 2-manifold)or in a volume (a 3-manifold).For instance,3-D thickening (Figure 2)builds a cylinder around each edge of the graph,and adds the necessary faces to connect these cylinders.Figure 2(a)shows a simple graph with only three edges,Figure 2(b)shows the surface obtained with 3-D thickening,Figure 2(c)shows a section of this object by a plane,and Figure 2(d)shows the successive sections of this*Correspondence to:D.Bechmann,LSIIT UPRES-A ULP-CNRS7005,UniversiteÂLouis-Pasteur Po Ãle API,Boulevard Se Âbastien Brant,F-67400Illkirch Cedex,France.THE JOURNAL OF VISUALIZATION AND COMPUTER ANIMATION put.Animat.2000;11:261±277..............................................................................................................................................................................................................................Copyright #2000John Wiley &Sons,Ltd.object,i.e.the actual animation.In this example,thesection of the 3-D object is a curve (a circle).The section of a 4-D object built with 4-D thickening is a surface,a topological sphere (Figure 3)or a topological torus.Figure 3(a)represents the object built by the 4-D thickening of the graph of Figure 2(a),projected on the xyt ,xzt ,xyz and yzt spaces.In Figure 3(b),this object is embedded in the xyz space,and the t coordinate is represented by a translation along the y axis.Fig-ure 3(c)represents the successive sections of this object,in the same way as the sequence in Figure 2(d).In this paper we will not describe the involved topological model in detail.The description of the thickening algorithm is abstract enough to be indepen-dent.We want a single model to manipulate both these 4-D objects and their sections,which are 3-D objects.To do this,we use generalized maps,1an n -D topological model.This model is homogeneous what-ever the dimension.Therefore,the generalization in n -D of the described algorithms is easy.The thickening method presented in this article describes mainly the construction of surfaces and volumes from the topological point of view,without support on the embedding of these objects.For this reason,the animations generated by the built objects comprise only evolutions of regular circles,spheresand toruses.However,by using other tools and operations,it is possible to modify the embedding of these objects after their construction with the thicken-ing operation.Thus it is possible to generate animation of objects which are homeomorphic to circles,spheres or toruses,but which have any arbitrary embedding.Figure 4shows a surface example in which a curve embedded in a maple leaf is separated into two leaves.We begin with the object built by 3-D thickening in Figure 2,in which we remove the boundaries,and we connect each one of its boundaries to the boundary of a new face (Figure 4b).The goal of these operations is to modify the embedding of the initial objects boundary.The only constraint in this process is that both boundaries must be topologically identical,i.e.with the same number of edges.Figure 4(c)shows the object obtained after modi®cation of its three boundaries,and Figure 4(d)shows the associated animation.It is also possible to modify the boundary of an object built with 4-D thickening by connecting it to a surface topologically identical to its boundary but with any arbitrary embedding.Another method to modify the shape of the animated objects consists in the use of tools for modi®cation of the embedding of the whole object,such as a deformation tool for instance.Figure 5presents the application of the deformationmodelFigure 1.An animation with topologicalmodi®cations.Figure 2.3-D thickening (b)of a simple graph (a),a section (c)and the associated animation (d).S.BRANDEL,D.BECHMANN AND Y.BERTRAND...............................................................................................................Visualization &Computer Animation...............................................................................................................Copyright #2000John Wiley &Sons,Ltd.put.Animat.2000;11:261±277262DOGME .2Figure 5(a)shows an object built with 4-D thickening with a toric section,and deformed with two constraints.The embedding of toruses in the associated animation (Figure 5b)is thus modi®ed;in particular,the animated objects can move away from the plane on which they move.In addition,a limitation of the thickening operation is that its use is not very intuitive.Indeed,itisFigure 3.4-D thickening (a and b)of a simple graph (Figure 2a)and the associated animation(c).Figure 4.Modi®cation of the embedding boundary of a surface.OPERATION FOR ANIMATION...............................................................................................................Visualization &Computer Animation...............................................................................................................Copyright #2000John Wiley &Sons,Ltd.put.Animat.2000;11:261±277263necessary to build a graph representing the trajectoryof the animated objects,and the construction of this graph is not always easy.A possible extension of this work consists in elaborating higher-level operations,`to transform n objects into p objects'for instance,using an underlying graph whose construction would be transparent for the user.Modi®cations of the embed-ding of the built objects could also be integrated.This would yield a catalogue of complex surface metamor-phoses in several others.Background and Related WorkSome methods allow building of animations compris-ing topological modi®cations.Let us only mention metamorphosis and the implicit surfaces,which are the techniques that bear the greatest resemblance space-time objects,the basis of this work.The morphing process of an object into another object (see Lazarus and Verroust 3for a complete state of the art in metamorphosis)can be taken as an animation.Numerous techniques de®ne the morphing between objects provided that they be topologically identical to spheres,4or topologically identical to each other.5Other techniques allow modif ication of the topology of the object during transformation.6,7In this objective,Bethel and Uselton 6carry out these transfor-mations by introducing degenerate grids,and Delin-gette et al .7de®ne basic operations on simple surfaces.One approach 8,9consists in de®ning intermediate surfaces which represent the precise moment of the topology change.In the method introduced by Cohen-Or et al .8,the intermediate objects are built by metamorphoses based on a distance ®eld in which the interpolation is controlled by a distortion function.We are particularly interested in the approach described by DeCarlo and Gallier,9which ®xes neither restric-tions nor particular conditions on the initial and ®nalobjects,since both objects are oriented triangulated surfaces.DeCarlo and Gallier 9study a method to grow holes in different ways,and apply their method to merge objects.This method requires a de®nition an intermediate object,which precisely marks the date of the topological modi®cation:the pinched sphere (the meeting of both poles)during the transformation of a sphere into a torus,for instance.Control meshes associated with the initial object and the ®nal object allow a precise control of the morph,but it is dif®cult to build animations with multiple topological modi®-cations.The evolution of an implicit surface constitutes also an animation.The application ®eld of the implict surfaces is extremely large:10pure modelling,geome-trical animation,physically based modelling.For example,an implicit function de®ning an iso-surface can be a distance with regard to a vertex or an axis.By combining several implicit functions with merging functions,we de®ne a distance with regard to a skeleton.Such an iso-surface corresponds to the thickening of a graph.One way of producing an animation is to dynamically modify the iso-value.11With such an animation,we can see the evolution of the thickness of the built object.When the the thickening of some edges of the skeleton meet,holes can be ®lled,thus avoiding self-intersections.More-over,one of the problems in the use of implicit surfaces is their visualization.Two methods are generally used:ray tracing 12and polyhedrical discretization.13The polygonization of the iso-surfaces is similar to the section of the space-time objects:a curve or a surface is built in both cases.But the result of the polygonization is not the same according to the chosen polygonization algorithm or the precision of the subdivision.Then the polygonization can generate inconsistencies.Finally,using space-time objects 14±18allows such mergings and splittings of curves or meshes.In this context,we can compute the section of a 4-D objectbyFigure 5.Deformation of a volume with toric section.S.BRANDEL,D.BECHMANN AND Y.BERTRAND...............................................................................................................Visualization &Computer Animation...............................................................................................................Copyright #2000John Wiley &Sons,Ltd.put.Animat.2000;11:261±277264a hyperplane,and an animation is the succession of these sections.Our context is the boundary representa-tion.In this domain,a4-D object is a3-manifold which represents in fact the boundary of this4-D object.In Aubert and Bechmann144-D objects are extrusions of 3-D objects;the shape of the4-D objects is under the control of a4-D deformation process.Another method to build4-D objects consists in computing the Carte-sian product of two objects.19This operation is the generalization of the extrusion operation.The type of the resulting animation depends on the type of operations used to create the4-D objects.For instance, the extrusion of a3-D sphere is inevitably a4-D sphere. In the section of this4-D sphere,a3-D object can be split into several3-D objects,but then these objects cannot be merged into a single object.Only toric objects allow these types of deformations.Therefore we have designed a4-D object modeller dedicated to animation17with a large set of operations,in order to increase the number of possible animations.These operations allow modelling of4-D objects with any shape and any topological structure. Thickening from a Surface The goal of this section is to detail the building principle of a surface(3-D thickening)or a volume(4-D thickening)from a graph drawn on a surface.The graph is created by picking edges on a surface.The3-D thickening is done in two steps:®rst we create a cylinder for each picked edge;next we close the nodes incident at one or more picked edges by inserting facets.In the same way,4-D thickening works in two steps:®rst we create a hypercylinder for each picked edge;next we insert volumes to close the nodes.For the sake of realism,we consider that the radius of the cylinders should remain constant,but the building of the cylinders uses the2-manifold on which the graph is drawn.Figure6(a)shows a graph drawn on an irregular2-manifold.The cylinders should remain regular whatever the embedding of the 2-manifold.For this reason,the distance between the boundary of the cylinder and the thickened edge is not expressed as being proportional to the lengths of the edges which are adjacent to graph edges(Figure6c). Figures6(b)and6(c)illustrate this case.In these ®gures,the cylinder boundaries are set in the middle of the edges which are incident to the graph.Therefore we have decided to always place the cylinder bound-ary at the same distance(called d)from the edges corresponding to the graph(Figure6d).In the®rst stage,distance d remains constant with each of the edges.The thickening with a radius associated with each picked edge is described later.3-D ThickeningEdge Thickening.The®rst step of the thickening from a graph drawn on a2-manifold consists in associating a cylinder with each picked edge.We begin by describing the construction of a single cylinder,when only one edge MP is picked.In the ®rst stage we suppose that both faces incident to MP are coplanar.These two faces de®ne two directions~i and~j(~i along edge MP,~j orthogonal to MP in the plane de®ned by both faces).The cylinder is built by setting four quadrilaterals around MP(Figure7a). Figure7(b)shows the distribution of the embeddings of one of the cylinder's boundaries.Two embeddings (M1and M2)are set on both edges(MQ and MR) which are adjacent to MP,at a distance d from MP. MP and MQ are incident to the same face,and so are MP and MR.Two other embeddings(M3and M4)are set on a straight line going through M and along a direction~k,on each side of M,and at a distance d from MP.The four other embeddings are computed with a similar method from point P.Points M1,M2,P1 and P2are always set on the edges that are adjacent to MP,which explains the shape of the cylinder'send Figure6.Thickening of a graph drawn on an irregular2-manifold.OPERATION FOR ANIMATION ............................................................................................................... Visualization&Computer Animation............................................................................................................... Copyright#2000John Wiley&Sons,put.Animat.2000;11:261±277265when the lattice is irregular (Figure 7c).These fourpoints are set at a distance d from edge MP ,which allows keeping the cylinder radius constant.When two cylinders are incident to the same vertex M ,we have isolated three cases to determine the cylinder adjacency (Figure 8).In Figure 8(a),both edges are incident to the same face.In this case,both cylinders necessarily intersect.Indeed,during the thickening of edge MP ,point M 2is set on edge MQ ,which is also thickened,and therefore set in the centre of a second cylinder.In Figure 8(b),both edges are incident to two distinct adjacent faces.M 2(resulting from the thickening of MP )and M '1(resulting from the thickening of MQ )are both set on MR ,generally identical.Then both cylinders are adjacent on two edges,which are M 3M 2and M 2M 4.Finally,in Figure 8(c),both edges are incident to two non-adjacent faces.The only contact point between bothcylinders is points M 3and M 4.When three or more edges are picked,treating these edges two by two enables one to always come down to one of these three cases.Both edges that are incident to a thickened edge MP are assumed coplanar in order to build a regular cylinder including MP (Figure 9a).Figure 9(b)repre-sents the building of a cylinder in the case where both faces are not coplanar.In order to convey a speci®c consistency to the built object,it is imperative that points M 3and M 4do not belong to the plane which is determined by the faces incident to MP .It is thereforenecessary that (~i ,~j ,~k )form a basis.That is the reasonwhy the initial 2-manifold must be planar.Never-theless,a suf®cient condition is to use a 2-manifold whose projection on a plane orthogonal to ~k presents no self-intersections.Hence,the 2-manifold is not necessarily planar.In order to have a uniformjunctionFigure 7.Building of the cylinder associated with an edge (a and c)and boundary of this cylinder(b).Figure 8.Different cases of the embedding of two cylinders incident to the same vertex.S.BRANDEL,D.BECHMANN AND Y.BERTRAND...............................................................................................................Visualization &Computer Animation...............................................................................................................Copyright #2000John Wiley &Sons,Ltd.put.Animat.2000;11:261±277266between the built cylinders,we decide to use the same direction ~k for all the edges (the direction ~k is normal to ~i and ~j ).We generally use a regular triangulated grid set on plane xt ,and a thickening direction along axis z .Node Closing.It is necessary to insert some faces inorder to connect the cylinders to each other at each node of the graph,and to manage the embedding of the vertices.Before closing the nodes,we modify the embedding of the cylinders,in order to avoid self-intersections,and to smooth the nodes.Figure 10describes the modi®cations done at the embeddings when we create two cylinders incident to a same vertex.In the case where two edges incident to the same face are picked for the thickening (see Figure 8a),we ®rst modify the embedding,in order to suppress the intersections between the cylinders.Let MP and MQ be these two edges (Figure 10a).M 2is created and set onthe bisecting line of MP and MQ ,at a distance d from both edges (Figure 10b).In all cases (Figures 8a±c),we bring other modi®ca-tions of the embedding to require face insertions at each node,in order to smooth the junction between two cylinders.M 3is divided into P 3and Q 3,M 4is divided into P 4and Q 4.Points P 3and P 4are shifted adistance d along direction MP !,and Q 3and Q 4alongdirection MQ !(Figure 10c).Each vertex M incident to at least one picked edge is closed by face insertion.These faces are the connec-tions between the edges incident to M .We insert connection facets corresponding to each pair edge (not necessarily picked)incident to the same face of the graph.Consequently,the face insertion algorithm works at the level of the faces incident to M in the 2-manifold supporting the graph,and not at the level of the edges.We have isolated three particular cases:let F be a face incident to M in the graph;both edges of F incident to M are picked (Figure 11a),only one of both edges of F incident to M is picked (Figures 11b andc),Figure 9.Cylinder associated with an edge whose incident faces are coplanar (a)or not(b).Figure 10.Modi®cation of the embedding of the cylinder at a node.OPERATION FOR ANIMATION...............................................................................................................Visualization &Computer Animation...............................................................................................................Copyright #2000John Wiley &Sons,Ltd.put.Animat.2000;11:261±277267and no edge of F is picked (Figure 11d).Depending oneach individual case,four faces (Figure 11a)or two faces (Figures 11b±d)are inserted in order to close the nodes.In Figures 11(a±d),point M 4which is symme-trical to M 3and the corresponding faces are not represented,in order to clarify the ®gures.When both edges MP and MQ incident to F are picked,point M 2is shared by both cylinders (Fig-ure 11a).Two quadrilaterals M 2P 3M 3Q 3and M 2P 4M 4M 4Q 4(the second not represented)are inserted in order to connect both cylinders.Because these two quadrilaterals are not coplanar,they are triangulated by inserting edges P 3Q 3and P 4Q 4.When only edge MP is picked (Figure 11b),two triangles M 2P 3M 3and M 2P 4M 4(the second not represented)are inserted.If,in addition,edge MR is picked (Figure 11c;see also Figure 8b),a second cylinder is built to thick MR .Therefore,a point M 1'is set on edge MQ .Here we suppose that M 2and M 1'are identical.On the contrary (M 2and M 1'separated),an additional triangle may be inserted.This treatment is similar to the management of the variable radius,which is detailed later.Finally,when none of the edges is picked (Fig-ure 11d),two triangles M 1M 2M 3and M 1M 2M 4(the second not represented)not connected to cylinders are inserted.P 1is set on segment MP Q 1is set on segment MQ ,both at a distance d from M .Figure 12(a)shows an example of a 3-D object obtained with this method,and Figure 12(b)shows the associated animation.4-D ThickeningThe interest of the method described in the previous section is to allow an easy extension to the superior dimension,in other words to generate 3-manifolds.The sections by hyperplanes of these objects are surfaces,and not only curves.In this section,these surfaces are topological spheres.The surface support-ing the graph has the same restriction as in 2-D thickening:it must be planar.This restriction has an important in¯uence on the resulting animation:the trajectory of the spheres'centres is a straight line.Edge Thickening.The ®rst step in 3-D thickening(see above)consists in associating a cylinder with each edge of the graph.In the same way,in4-DFigure 11.Faces insertion at the level of a graphnode.Figure 12.Thickened graph (a)and associated animation (b).S.BRANDEL,D.BECHMANN AND Y.BERTRAND...............................................................................................................Visualization &Computer Animation...............................................................................................................Copyright #2000John Wiley &Sons,Ltd.put.Animat.2000;11:261±277268thickening we associate a hypercylinder with each picked edge.Such a hypercylinder comprises eight prisms.In fact,in3-D thickening,two points M3and M4correspond to a point M1which belongs to an edge adjacent to a picked edge(Figure7b).These three points form the basis of two quadrilaterals.In the same way,4-D thickening associates four points with any point M1.These four points(taken two by two)form with M1the triangular basis of four prisms. Figure13(a)reminds the construction of bothquadrilaterals of half a2-D cylinder(see above). Figure13(b)represents the same object from earlier another point of view.In the front view,each of both faces looks like a simple edge(M1M3and M1M4). These edges represent in fact the quadrilateral's basis. We have seen that M1is set on the straight line MQ at a distance d from MP,M3and M4are set on a straight line d1going through M and along a direction~k,on each side of M,and at a distance d from MP.In the same way that Figure13(b)shows the basis of two quadrilaterals,Figure13(c)shows the basis of four prisms,which form the half hypercylinder.Each of these bases is formed®rstly by point M1,secondly by a point set on the straight line d1going through M and along~k,and thirdly by a point set on the straight line d2going through M and along~l.M3and M4are set on d1on each side of M,and M5and M6are set on d2on each side of M,all four at a distance d from MP. Thus we can form four distinct triangles,which are the basis of four prisms constituting the half hypercylinder.We proceed similarly to the second half hypercylinder,by building four prisms from point M2,which is symmetrical to M1with regard to M.Figure14(a)shows a cylinder composed of four quadrilaterals(in®ne lines),a section(in thick lines) by a plane and this cutting plane.Figure14(b)represents the projection in the space of a hypercylinder(in®ne lines)and a section(in solid surfaces)by a hyperplane.Node Closing.As in3-D thickening,in order to avoid intersections between hypercylinders and to smooth the junction between hypercylinders it is necessary to modify a certain number of embeddings. This operation is carried out before volume insertion to close the nodes.Figure15represents the successive operations.In order to clarify the®gure,we represent only one of the eight prisms forming a hypercylinder. It concerns the prism whose basis is M2M3M5(see Figure13c).The treatment is similar for the prisms whose bases are M2M3M6,M2M4M5,M2M4M6,and the prisms corresponding to point M1.Two hypercylinders intersect when two edges incident to the same face are picked.A®rst treatment is carried out in this case.Figure15(a)represents two such edges,when only one of the eight prisms normally composing a hypercylinder is represented for each edge.These two prisms share points M3and M5.M2is created and set on the bisecting line of MP and MQ,at a distance d from both edges.M2becomes a third vertex shared by both prisms(Figure15b).Figure13.Construction of a hypercylinder associated with anedge.Figure14.Cylinder(a)and hypercylinder(b)with theirsections.OPERATION FOR ANIMATION ............................................................................................................... Visualization&Computer Animation............................................................................................................... Copyright#2000John Wiley&Sons,put.Animat.2000;11:261±277269。
Aspects of Gravitational Clustering
ASPECTS OF GRAVITATIONAL CLUSTERING
3
ˆk is a linear second order differmode, labeled by a wave vector k. Here L ential operator in time. Solving this set of ordinary differential equations, with given initial conditions, we can determine the evolution of each mode separately. [Similar procedure, of course, works for the case with Ω = 1. In this case, the mode functions will be more complicated than the plane waves; but, with a suitable choice of orthonormal functions, we can obtain a similar set of equations]. This solves the problem of linear gravitational clustering completely. There is, however, one major conceptual difficulty in interpreting the results of this program. In general relativity, the form (and numerical value) of the metric coefficients gαβ (or the stress-tensor components Tαβ ) can be changed by a relabeling of coordinates xα → xα′ . By such a trivial change we can make a small δTαβ large or even generate a component which was originally absent. Thus the perturbations may grow at different rates − or even decay − when we relabel coordinates. It is necessary to tackle this ambiguity before we can meaningfully talk about the growth of inhomogeneities. There are two different approaches to handling such difficulties in general relativity. The first method is to resolve the problem by force: We may choose a particular coordinate system and compute everything in that coordinate system. If the coordinate system is physically well motivated, then the quantities computed in that system can be interpreted easily; 0 to be the perturbed mass (energy) density for example, we will treat δT0 even though it is coordinate dependent. The difficulty with this method is that one cannot fix the gauge completely by simple physical arguments; the residual gauge ambiguities do create some problems. The second approach is to construct quantities − linear combinations of various perturbed physical variables − which are scalars under coordinate transformations. [see eg. the contribution by Brandenberger to this volume and references cited therein] Einstein’s equations are then rewritten as equations for these gauge invariant quantities. This approach, of course, is manifestly gauge invariant from start to finish. However, it is more complicated than the first one; besides, the gauge invariant objects do not, in general, possess any simple physical interpretation. In these lectures, we shall be mainly concerned with the first approach. Since the gauge ambiguity is a purely general relativistic effect, it is necessary to determine when such effects are significant. The effects due to the curvature of space-time will be important at length scales bigger than (or comparable to) the Hubble radius, defined as dH (t) ≡ (a/a ˙ )−1 . Writing
Monocular Model-Based 3D Tracking of Rigid Objects
implementing a 3D tracking system and a discussion of the future of vision-based 3D tracking. Because it encompasses many computer vision techniques from lowlevel vision to 3D geometry and includes a comprehensive study of the massive literature on the subject, this survey should be the handbook of the student, the researcher, or the engineer who wants to implement a 3D tracking system.
Many potential Augmented Reality (AR) applications have been explored, such as medical visualization, maintenance and repair, annotation, entertainment, aircraft navigation and targeting. They all involve superposing computer generated images on real scenes, which must be done at frame-rate in online systems. 3D real-time tracking is
1
Introduction
Tracking an object in a video sequence means continuously identifying its location when either the object or the camera are moving. There are a variety of approaches, depending on the type of object, the degrees of freedom of the object and the camera, and the target application. 2D tracking typically aims at following the image projection of objects or parts of objects whose 3D displacement results in a motion that can be modeled as a 2D transformation. An adaptive model is then required to handle appearance changes due to perspective effects or to deformation. It can provide the object’s image position in terms of its centroid and scale or of an affine transformation [141, 26, 62]. Alternatively, more sophisticated models such as splines [16], deformable templates [142], 2D deformable meshes [112], or 2D articulated models [20] can be used. However, none of these methods involves recovering the actual position in space. By contrast, 3D tracking aims at continuously recovering all six degrees of freedom that define the camera position and orientation relative to the scene, or, equivalently, the 3D displacement of an object relative to the camera.
翻译(赵宏伟)
模型试验动态施工力学效应大跨度岩土隧道©上海交通大学和斯普林格出版社柏林海德堡2011年版文摘:大时间一直是一个争论的焦点为大跨度岩土隧道施工方法。
合理的施工方法对隧道的稳定性和施工进度有很大影响。
变形和失败的围岩是相当复杂的。
关联的大跨度岩土隧道郑州-在中国西安高速客运铁路线,大型模型试验与等比1:20研究在各种施工方法的动态力学行为。
它们包括正面的挖掘支持和不支持,和钳工加工方法的支持。
发现预变形和压力积累发生在工作面。
三种施工方法是进一步研究的影响,尤是在隧道位移和应力变化。
透露,钳工加工方法转移负载unexcavated区域,限制水平变形,有效地减少了应力集中,延长了峰值的位置之间的距离对于应力集中和工作面,从而增加稳定性。
模型试验结果不仅为确定合理的施工方法,提供理论基础但也可以作为参考类似的隧道和地下工程施工。
关键词:物理模型、施工方法、岩土隧道施工力学、稳定CLC号码:你452.2文档代码:0引入大型铁路的快速发展建筑在中国的脸和工作隧道的跨度越来越大。
的关键大跨度隧道的成功一直consid -感染一种高度复杂的问题的观点工程,在很大程度上减少经验公式(1 - 2)提供了良好的结果类似的地质条件。
隧道的建设诱发压力变化对地面与对应荷兰国际集团(ing)位移。
如果采用不当,它会导致更高的工程建设的风险,在-折痕成本,过度的围岩破坏区,甚至隧道崩溃。
演唱和胫骨[3]研究了在支持时间根据新的软岩隧道奥地利隧道的方法。
大量的研究工作完成隧道围岩级别和供给接口技术[4 - 7]。
尽管一些以往的研究进行了失效机理,动态的大跨度岩土隧道和压力的行为工作面与尊重变化——很少研究我们的施工方法(8 - 12)。
通Luo-chuan隧道线路在高的郑州-西安线路高速铁路客运线是两车道的岩土,这是控制工程之一。
工作的区域脸是164平方米,跨度为15.2米。
由于大型跨越,建设的难度大幅增加。
作为由于扰动围岩的构造-,合理的施工方法有影响稳定和工程成本。
sci处理流程待定英语
sci处理流程待定英语English: The SCI (Science Citation Index) processing workflow is a crucial stage in the academic publishing process. This workflow typically involves several steps, starting from the submission of a manuscript to a scientific journal to its final indexing in the SCI database. The first step involves the submission of the manuscript to a journal, where it undergoes a peer review process to ensure its quality and relevance to the field. Once accepted, the manuscript goes through editing and formatting to meet the journal's guidelines. Subsequently, the journal submits the metadata of the article to the SCI for indexing. The SCI team checks the metadata for accuracy and relevance before including it in their database. Once indexed, the article becomes part of the SCI database, making it accessible to researchers worldwide. The SCI processing workflow is an essential part of the scholarly communication ecosystem, ensuring that high-quality research is disseminated efficiently and effectively.中文翻译:SCI（科学引文索引）处理流程是学术出版过程中至关重要的一个阶段。
On the Control of Space Free-Flyers Using Multiple Impedance Control
Proceedings of the IEEE International Conference on Robotics and Automation, April 21-27 1997, Albuquerque, NM.On the Control of Space Free-Flyers UsingMultiple Impedance ControlS. Ali A. Moosavian, Evangelos PapadopoulosDepartment of Mechanical Engineering & Centre for Intelligent MachinesMcGill UniversityMontreal, Quebec, Canada H3A 2A7Abstract. The Multiple Impedance Control (MIC) is a new algorithm which enforces a designated impedance on both a manipulated object, and all cooperating manipula-tors.In this paper, the MIC is applied to a space robotic system in which robotic arms, mounted on a free-flying-base, manipulate an object. The general formulation of the MIC is extended to include the dynamic coupling between the arms and the base. It is shown that under the MIC law, all participating manipulators, the free-flyer base, and the manipulated object exhibit the same designated impedance behavior. This guarantees good tracking of system manipulators and the object, in performing a manipulation task. A system of two cooperating two-link manipulators is simulated, in which a Remote Centre Compliance is attached to the second end-effector. The object is grabbed with a pivoted grasp condition, i.e. both the translational and rotational motions of the object have to be controlled by end-effector forces. As simulation results show, the re-sponse of the MIC algorithm is smooth, even in the occur-rence of an impact due to collision with an obstacle.I. Introduction.Free-flying space manipulator systems, in which robotic manipulators are mounted on a free-flying spacecraft, are envisioned for assembling, maintenance, repair, and contingency operations in space. Early research work in this area focused on the dynamics and motion control of a single manipulator in free-floating mode [1]-[4], i.e. an end-effector moves toward a target in the inertial or spacecraft body-fixed frame with no significant force interactions between the system and the environment. Dynamics and motion control of multiple manipulators in both free-floating and free-flying modes have been studied by various researchers recently [5]-[7]. However, coordination and control of the spacecraft and its multiple manipulators during capture or manipulation of objects has not attracted adequate attention. These tasks require employing force or impedance control strategies, so that interaction forces and system response during contact are controlled.As an extension of Hogan’s impedance control concept [8], the Object Impedance Control (OIC) has been developed for multiple robotic arms manipulating a common object [9]. A combination of feedforward and feedback control is employed to make the object behave like a reference impedance. However, it has been realized that applying the OIC to manipulation of a flexible object may lead to instability [10]. Based on the analysis of a representative system, it was suggested that in order to solve the instability problem, one should either increase the desired mass parameters or filter and lower the frequency content of the estimated contact force.In a recent study, a new algorithm named as Multiple Impedance Control (MIC) was developed which enforces a designated impedance of both manipulator end-points, and of a manipulated object [11]. Physically speaking, this means that all participating end-effectors and the manipulated object are controlled to behave like a designated impedance in reaction to any disturbing external force on the object. This results in good tracking of the various manipulators of the system and the object. The MIC algorithm is able to perform both free motions and contact tasks without switching between control modes. In addition, object inertia effects are compensated for in the impedance law, and the end-effector(s) tracking errors are controlled.In this paper, the new MIC algorithm is applied to space robotic systems in which manipulators are mounted on a free-flying base. The general formulation is adapted to consider the dynamic coupling between the arms and the base while the manipulated object may include an internal source of angular momentum. Next, it is shown by erroranalysis that under the MIC law all participating manipulators, the free-flyer base, and the manipulated object exhibit the same designated impedance behavior.Finally, a system of two cooperating tw o-link manipulators is simulated, and the obtained results are discussed.II. The MIC Law for Space Free-Flyers.When applied to a terrestrial system, the MIC strategy enforces the sa me impedance relationship at the manipulator end-effector level, and at the manipulated object level. In space, since the cooperating robotic arms are connected through a free-flying base, the MIC algorithm is applied so that all participating manipulators, the spacecraft, and the manipulated object exhibit the same impedance behavior,as implied by "multiple" in naming the MIC. This strategy allows coordinated motion/force control of the space free-flying robot for performing a manipulation task. In this section, following a brief review on space free-flyers and object dynamics, the MIC law for space applications is presented.(a) System Dynamics Modelling. The vector of generalized coordinates for a space free-flyer with multiple manipulators, shown in Figure 1, can be chosen asq R =(,,)C T T T T0d q (1)where R C 0describes the inertial position of the spacecraft center of mass (CM), d 0 is a set of Euler angles that describes the orientation of the spacecraft, and q = q q q ()()(),,,12T T n T TL ()is a K ´1 column vector which contains all joint angle vectors. The q ()m is an N m ´1column vector which contains the joint angles of the m-thmanipulator, and K N ==åm m n1. Assuming that the systemconsists of rigid elements and applying the general Lagrangian formulation, the equations of motion can be obtained as [12]-[13]Denotesbodycenter ofFig. 1: A space free-flyer with n manipulators.Hq C Q (,)˙˙(,˙,,˙)(,)d q d d q q d q 0000+=(2)where H is an N ´N positive definite mass matrix of the system (N =K +6 is the total system degree of freedom),C is an N ´1 vector which contains all the nonlinear velocity terms (in a microgravity environment), and Q is the N ´1 vector of generalized forces.The vector of output (controlled) variables is defined as˜[,,,,,,]()()()()x R x x =C T T E T E T E n T E n T T011d d d L (3)where x E m () describes the m-th end-effector inertialposition, and d E m ()is a set of Euler angles which describes the m-th end-effector orientation. It is assumed that all manipulators have six DOF, i.e. K =6n (n is the number of participating manipulators), and that they all participatein manipulating the object. The vector of output speeds ˜˙xis obtained from the time derivative of the generalizedcoordinates (˙q), using a square Jacobian J C ˜˙˙x J q =C(4)The equations of motion can now be written in thetask space, i.e. in terms of the output coordinates ˜x, as ˜()˜˙˙˜(,˙)˜H q x C q q Q +=(5a)where˜H J H J =--C T C1˜˜˙˙C J C H J q =--C T C˜Q J Q =-CT (5b)To develop the MIC law, the vector of generalizedforces in the task space, ˜Q, is written as ˜˜˜˜˜˜Q Q Q Q Q Q =+=++appreactmfreact(6)where ˜Q react is the reaction force on the end-effectors, and ˜Q appis the applied controlling force consisting of the force which corresponds to the motion of the system, ˜Q m, and of the required force to be applied on the manipulatedobject by the end-effectors, ˜Q f. These terms will be detailed after describing object dynamics.(b) Object Dynamics. The equations of motion for the object can be written based on rigid-body dynamics.For a flexible object an appropriate dynamics model can be simply substituted for the following model. Also, the object may include an internal angular momentum source,see Figure 2. Thus, the object dynamics can be expressed asMx F F F GF ˙˙+=++w c o e (7)where M is the mass matrix, x x =(,)G T obj T Td describes the position of the object center of mass x G and the object orientation described by Euler angles d obj , F w is a vector of nonlinear velocity terms, F c describes the contact forces/moments, F o describes external forces/torques (other than contact and end-effector ones), F e is a 6n ´1 vectorwhich contains all end-effector forces/torques applied on the object (F e i () is a 6´1 vector corresponding to the i-th end-effector), and the matrix G is referred to as the grasp matrix, [11]. Next, using the system dynamics model and the object dynamics equations, the MIC law for space applications is developed.(c) The Control Law. A desired impedance law for the object motion can be chosen asM e k e k e F 0des d p c ˙˙˙+++=(8)where e x x =-()des describes the object tracking error, k pand k d are control gain matrices, and M des is the object desired mass matrix. Then, by direct comparison of Eq. (8)and Eq. (7), it can be seen that the desired impedance behavior can be obtained ifGF MM M xk e k e F F F F e des des des d p c c o req=+++()+-+()-1˙˙˙w (9)provided that the matrix S obj which relates the objectangular velocity, w obj , to the Euler rates, ˙d obj, as [14] w d obj obj obj=S ˙(10)is not singular. Clearly, this depends on the Euler angles definition. Therefore , applying the required end-effector forces/torques on the object, F e req, results in the targeted impedance relationship as described in Eq. (8). Eq. (9) can be solved to obtain a minimum norm solution, resulting inF G MM M x k e k e F F F F e des des des d p cc oreq=+++()+-+()-#{˙˙˙ˆˆ}1w (11)where G # is the pseudoinverse of the grasp matrix, a full-rank matrix (provided that S obj is not singular) defined asG W G GW G #=()---111T T(12)weighted by a task weighting matrix W , so that linear andangular motions or their components are weightedappropriately. Note that ˆF cis the estimated value of the contact force F c which can be computed as [11]ˆˆ˙˙F Mx F F GF coe=+--w(13)f o n o,o (16)to the derivation for ˜Q fand assuming that the system mass and geometric parameters are known, ˜Q mcan be obtained as˜˜˜˜˜˙˙˜˜˙˜˜ˆ˜Q H M M x k e k e U F C mdesdes des d pf cc=+++[]+-1(19)where ˜M des -1 is the block-inverse of ˜M des .III. Error Analysis.Substituting Eqs. (19), (17a), and (16) into Eq. (6), andthe result into Eq. (5a) yields˜˜˜˜˙˙˜˜˙˜˜˜˙˙˙˙˙˙˙#H M M x k e k e U F x0G M M M x k e k e F x 0desdes des d pf cdes des des d p c c-´-+++()-()++++()-()ìíïîïüýïþï=1611(20)where it is assumed that the exact value of the contact forceis available, and that the mass and geometric properties for the manipulated object, and the space free-flying manipulator system are known. Since Eq. (20) must holdfor any M and any ˜H, it is concluded that ˜˜˜˜˙˙˜˜˙˜˜˜˙˙˙˙˙˙˙#H M M x k e k e U F x 0G M M M x k e k e F x0des des des d p f c des des des d p c c--+++()-()=+++()-()=11(21)Since G # is of full-rank, and M and ˜Hare positive definite inertia matrices, Eq. (21) results in˜˜˙˙˜˜˙˜˜˙˙˙M e k e k e U F 0M e k e k e F 0des d p f c des d p c c+++=+++=(22)Considering the definitions for ˜Mdes , ˜k d , ˜k p, and U f c, Eq. (22) means that all participating manipulators,the free-flyer-base, and the manipulated object exhibit the same impedance behavior. This guarantees an accordant motion of the various subsystems during object manipulation tasks.IV. Simulation Results.Task Definition. Figure 3 shows a robotic system inplanar motion, performing a cooperative manipulation task ,i.e. moving an object with two manipulators according to predefined trajectories. It is assumed that the position and attitude of the system base is controlled and does not move. One of the two end-effectors is equipped with a Remote Centre Compliance (RCC). The task is to move an object based on a given trajectory which for illustration purposes passes through an obstacle. The object has to come to a smooth stop at the obstacle. Initially, the object has been grabbed with a pivoted grasp condition, i.e. no torque can be exerted on the object by the two end-effectors. Therefore, both the translational and rotational motions of the object are controlled by end-effector forces .Simulation Results and Discussions. For thesystem depicted in Figure 3, the geometric parameters,mass properties, and the maximum available actuator torques are displayed in Table 1. The origin of the inertial frame is considered to be located at joint 1 of the first manipulator, and joint 1 of the second manipulator is at (1.2 m, 0.0)T . The object and controller parameters are m kg I kg m m obj G e e ===-=-()3005030020102.,.,.,.()()r r M k k des p d diag diag diag ===(,),(,),(,)1010100100300300The initial conditions are (,,˙,˙,,,˙,˙,˙)(.,.,,,.,.,,,,)(,/)()()()()()()()()q q q q q q q q rad rad s T T 112111211222122227270010250000q q,=-It is assumed that the RCC unit is initially free of tension or compression, where its stiffness and damping properties are chosen as k e diag kg =´(,)/sec 221042, and b e diag =(,55102)/sec ´kg , see [15].The desired trajectory for the object center of mass,expressed in the inertial frame, isx e m y m G des t G des des =-==-1050,.,q q Table 1: The system Parameters.Mani-pulator i-th body i r i(m)(m)i l i(m)(m)m i (m)(kg)I i (m)(kgm 2)t i (m)(N-m)110,0.500,-0.5010.0 1.50100.0120,0.500,-0.50 6.00.80100.0210,0.500,-0.5010.0 1.50100.0220,0.500,-0.508.00.80100.0)q 2(2)Fig. 3:Two robotic arms mounted on aspacecraft, performing a cooperative manipulation task on a plane.where q 0 describes the object initial orientation. The obstacle is at x m w =12., so it is expected that the objectwill come in contact at its right side, i.e. at x r G e +()2. It is assumed that no torque is developed at the contact surface (i.e. a point contact occurs), therefore n c is equal to the moment of f c . Also, there is no other external force applied on the object, i.e. f 0n 0o o ==,. The contact force is estimated based on Eqs. (13, 14b), where the real stiffness of the obstacle is k e N m w =15/. The time step,D t , in the estimation procedure (Eq. (14)) is 10 msec.Given the above information, the obtained simulation results are presented in Figure 4.As shown in Figures 4a,b the y-component of the error in the object position, starting from some initial value,converges to zero smoothly. This is due to the fact that contact occurs along the x-direction, and so the contact force does not affect the object’s motion in the y-direction.The x-component of the error, decreases at some rate until contact occurs at t »10. sec. This rate changes after contact, because the error dynamics depend on the dynamics of the environment, according to the impedance law. Then, this error smoothly converges to the distance between the final desired x-position and the obstacle x-position.The object orientation error, starting from zero, grows to some amount and then converges to zero, Figure 4a. The initial growth is due to the fact that the first end-effector (i.e. without the RCC unit) responds faster than the second one which is equipped with the RCC. Therefore, the difference between the two end-effector forces produces some moments which results in an undesirable rotation of the object. However, after a short transient period the difference vanishes and so does the object orientation error.Actuator saturation limits are reached at start-up (because of large initial errors and error-rates), and at the time of contact, Figures 4c,d. Joint torques for the first manipulator converge to a steady state soon after contact (about half of a second), while it takes longer for those of the second manipulator. Again, this is due to the existence of the RCC.The contact with the obstacle occurs along the x-direction when the right end of the object goes beyond x w .Therefore, f c yremains equal to zero before and after contact, while f c xappears whenever the object is in contact with the obstacle, Figure 4e. As the impact energy is dissipated, f c xconverges to a constant value. According to the imposed impedance law, Eq. (8), for diagonal gain matrices this constant force has to be equal to -=-=k e p x 10001(.) -10N , which is verified from the response results. Figure 4f shows the difference between the real value of the contact force, and the estimated one used by the controller. As can be seen, the difference isalmost zero except during a very short period following impact. Even then, the difference is quite small (about 10%of the real value). After this period, the acceleration profiles become smoother and the difference between the real and estimated values of the contact force becomes zero. Note that before the contact, the slight difference between the two is due to the approximation of object acceleration,based on calculation of Eq. (14).0051015P o s . & O r i e n . E r r o r (m , r a d )Time (sec)-0.6-0.4-0.200.20.40.60.81V e l o c i t y E r r o r (m /s , r a d /s )Time (sec)(a)(b)-100-500501000.511.521-s t A r m J o i n t T o r q u e s (N .m )Time (sec)-100-5050100012345672-n d A r m J o i n t T o r q u e s (N .m )Time (sec)(c)(d)-240-200-160-120-80-40001234567C o n t a c t F o r c e (N )Time (sec)-30-20-100102000.51 1.52R e a l-E s t i m a t e d C o n t a c t F o r c e (N )Time (sec)(e)(f)Fig. 4:Simulation results, (a) Object trackingerrors, (b) Velocity errors, (c) Manipu-lator 1 joint torques, (d) Manipulator 2joint torques, (e) Contact force, F c (real value), (f) Difference between the real and estimated values of contact force.A comparative analysis between existing control strategies reveals that use of a standard impedance law does not provide compensation for the object's inertia forces and yields unacceptable results when the object is massive, or when it experiences large accelerations [11]. Also, the OIC which implements the impedance law at the object level, is basically formulated for a system with rigid elements, and does not yield a good tracking in the presence of system flexibility. The more flexible the object is, the worse the performance of the OIC will be. On the other hand, as shown by simulation, performance of the MIC algorithm applied to a cooperative manipulation task is excellent,even in the presence of flexibility, and during impact with an obstacle.V. Conclusions.In this paper, the new Multiple Impedance Control (MIC) was developed and applied to a space robotic system. The MIC enforces a designated impedance on cooperating manipulators and on the manipulated object, which results in a harmonious motion of various subsystems. Similar to the standard impedance control, one of the benefits of this algorithm is the ability to perform both free motions and contact tasks without switching the control modes. In addition, an object's inertia effects are compensated for, in the impedance law, and at the same time the end-effector(s) tracking errors are controlled. To consider the dynamic coupling between the arms and the base in space, the general MIC formulation was expanded. By error analysis it was shown that, under the MIC law, all participating manipulators, the free-flyer base, and the manipulated object exhibit the same designated impedance behavior; resulting in an adjusted tracking of various manipulators of the system together with the object. It was shown by simulation that even in the presence of flexibility and impact forces, the MIC yields a smooth and stable performance.VI. Acknowledgments.The support of this work by the Natural Sciences and Engineering Council of Canada (NSERC) is acknowledged. We would also like to acknowledge support of the first author from the Iran Ministry of Higher Education.References[1]Vafa, Z. and Dubowsky, S., “On The Dynamics ofManipulators in Space Using The VirtualManipulator Approach,” Proc. of IEEE Int. Conf. onRobotics and Automation, April 1987, pp. 579-585.[2]Umetani, Y. and Yoshida, K., “Resolved MotionRate Control of Space Manipulators withGeneralized Jacobian Matrix,” IEEE Transactions onRobotics and Automation, Vol. 5, No. 3, June1989, pp. 303-314.[3]Alexander, H. and Cannon, R., “An ExtendedOperational-Space Control Algorithm for SatelliteManipulators,” The Journal of the AstronauticalSciences, Vol. 38, No. 4, October-December 1990,pp. 473-486.[4]Papadopoulos, E. and Dubowsky, S., “On TheNature of Control Algorithms for Free-FloatingSpace Manipulators,” IEEE Transactions onRobotics and Automation, Vol. 7, No. 6, December1991a, pp. 750-758.[5]Yoshida, K., Kurazume, R., and Umetani, Y.,“Dual Arm Coordination in Space Free-FlyingRobot,” Proc. of IEEE Int. Conf. on Robotics andAutomation, April 1991, pp. 2516-2521.[6]Dubowsky, S. and Papadopoulos, E., “TheDynamics and Control of Space Robotic Systems,”IEEE Transactions on Robotics and Automation,Vol. 9, No. 5, October 1993, pp. 531-543.[7]Papadopoulos, E. and Moosavian, S. Ali A., "AComparison of Motion Control Algorithms forSpace Free-flyers," Proc. of the 5th Int. Conf. onAdaptive Structures, Sendai, Japan, December 5-7,1994c.[8]Hogan, N., “Impedance Control: An Approach toManipulation -A Three Part Paper,” ASME Journalof Dynamic Systems, Measurement, and Control,Vol. 107, March 1985, pp. 1-24.[9]Schneider, S. A. and Cannon, R. H., “ObjectImpedance Control for Cooperative Manipulation:Theory and Experimental Results,” IEEETransactions on Robotics and Automation, Vol. 8,No. 3, June 1992, pp. 383-394.[10]Meer, D. W. and Rock, S. M., “Coupled-SystemStability of Flexible-Object Impedance Control,” inProc. of the IEEE Int. Conf. on Robotics andAutomation, Nagoya, Japan, May 1995, pp. 1839-1845.[11]Moosavian, S. Ali A., "Dynamics and Control ofFree-Flying Manipulators Capturing Space Objects,"Ph.D. thesis, McGill University, Montreal, Canada,June 1996.[12]Papadopoulos, E. and Moosavian, S. Ali A.,“Dynamics & Control of Multi-arm Space RobotsDuring Chase & Capture Operations,” Proc. Int.Conf. on Intelligent Robots and Systems (IROS‘94), Munich, Germany, Sept. 12-16, 1994a. [13]Papadopoulos, E. and Moosavian, S. Ali A.,“Dynamics & Control of Space Free-Flyers withMultiple Arms,” Journal of Advanced Robotics,Vol. 9, No. 6, 1995, pp. 603-624.[14]Meirovitch, L., Methods of Analytical Dynamics,McGraw-Hill, 1970.[15]De Fazio, T. L., Seltzer, D. S., and Whitney, D. E.,“The Instrumented Remote Centre Compliance,”Journal of The Industrial Robot, Vol. 11, No. 4,December 1984, pp. 238-242.。
toward a science of translating = 翻译科学探索
toward a science of translating = 翻译科学探索Toward a Science of TranslatingTranslation is an important part of communication between people of different languages. It is a complex process that requires a deep understanding of both the source language and the target language. As the world becomes increasingly interconnected, the need for accurate and reliable translation services is growing. In order to meet this need, it is important to develop a science of translating.The first step in developing a science of translating is to understand the different types of translation. There are two main types of translation: literal translation and interpretive translation. Literal translation is a direct translation of the source language into the target language, while interpretive translation is a more creative approach that takes into account the cultural context of the source language.The second step is to understand the different methods of translation. There are two main methods of translation: machine translation and human translation. Machine translation is a process in which a computer program is used to translate text from one language to another. Human translation is a process in which a human translator is used to translate text from one language to another.The third step is to understand the different techniques used in translation. There are several techniques used in translation, including literal translation, interpretive translation, and cultural adaptation. Literal translation is a direct translation of the source language into the target language. Interpretive translation is a more creative approach that takes into account the cultural context of the source language. Cultural adaptation is a process in which the translator adapts the source language to the target language in order to make it more understandable to the target audience.The fourth step is to understand the different tools used in translation. There are several tools used in translation, including dictionaries, translation software, and machine translation. Dictionaries are used to look up words and phrases in the source language and their equivalentsin the target language. Translation software is used to automate the process of translating text from one language to another. Machine translation is a process in which a computer program is used to translate text from one language to another.The fifth step is to understand the different approaches used in translation. There are several approaches used in translation, including literal translation, interpretive translation, and cultural adaptation. Literal translation is a direct translation of the source language into the target language. Interpretive translation is a more creative approach that takes into account the cultural context of the source language. Cultural adaptation is a process in which the translator adapts the source language to the target language in order to make it more understandable to the target audience.The sixth step is to understand the different theories used in translation. There are several theories used in translation, including the Sapir-Whorf hypothesis, the source-oriented approach, and the target-oriented approach. The Sapir-Whorf hypothesis states that language shapes thought and that translation is a process of transferring meaning from one language to another. The source-oriented approach is a process in which the translator focuses on the source language and attempts to accurately translate it into the target language. The target-oriented approach is a process in which the translator focuses on the target language and attempts to accurately translate it into the source language.The seventh step is to understand the different technologies used in translation. There are several technologies used in translation, including machine translation, natural language processing, and computer-assisted translation. Machine translation is a process in which a computer program is used to translate text from one language to another. Natural language processing is a process in which a computer program is used to analyze and understand natural language. Computer-assisted translation is a process in which a computer program is used to assist the translator in the translation process.By understanding the different types, methods, techniques, tools, approaches, theories, and technologies used in translation, we can begin to develop a science of translating. This science will help us to better understand the complexities of translation and to develop more accurate and reliable translation services.。
跨火Mil Dot 瞄准圈用户指南说明书
Crossfire Mil Dot ReticleWelcome to the Crossfire Mil Dot reticle user’s guide. This reticle allows the user to do range estimations as well as estimating hold-over, wind drift and lead moving targets. It is very important to understand that in order to use thesefeatures, the scope must be set at 14x magnification. Of course, the standard center crosshair can always be used at any magnification.Vortex®2Mil Dot:The term ‘mil’ in a mil dot reticle refers to a milliradian. A milliradian is a fraction of a circle, similar in concept to a degree. The dot spacing used in the Mil Dot reticle will correspond to 3.41 minutes of angle.Remembering that 1 minute of angle always equals 1.047 inches at 100 yards, we then know the dot spacing will be 3.6 inches at 100 yards.The following diagram illustrates the dot spacing on the reticle. Vertical dimensions are the same as horizontal .®3100 yards ---------------------- 3.6”200 yards ---------------------- 7.2”300 yards --------------------- 10.8”400 yards --------------------- 14.4”500 yards ------------------------ 18”600 yards --------------------- 21.6”700 yards -------------------- 25.2”800 yards -------------------- 28.8”Mil Width at Distances:4Ranging Distance:To use a Mil Dot for ranging purposes, you must have an object of known dimension at the same distance as your target to compare the mil spacing to.Examples of known dimensions:• A fence known to be 36” tall next to the coyote you’re shooting at.• The brisket-to-back distance on a whitetail buck of 18”.• The height of a standing ground hog of 10”.• A target 20” in diameter.Using our first example, we place our reticle on the fence with the horizontal crosshair even with the ground ( Remember that the scope must be turned to 14X). Reading our mils, we see that the fence equals 2 mils in height.5Using a simple formula, we now can calculate the distance to the fence (and the coyote) at 500 yards. This formula will be used for all ranging situations._____________1 yard (36”) x 10002500 yards to fence & coyote=______________________Known Dimension (in yards) X 1000Mils Read= Yards to Target6Windage compensation:Using your Mil Dot reticle for windage and moving target leads will require thorough knowledge of your cartridges ballistic performance and experience in properly reading wind strengths. Again, the scope must be at 14X for this to work.Remembering that 1 mil equals 3.6” at 100 yards, 7.2” at 200 yards, 10.8” at 300 yards etc., use the mil dots on the horizontal crosshair to hold-off the required distance. Remember to hold into the wind direction when doing this.For example, lets say you’re shooting at a target 400 yards away, and through experience and ballistics charts you believe the bullet will wind-drift about 7”. At 400 yards, the mil spacing equals 14.4”, so you know that you’ll need to hold about ½ mil into the wind to make your shot.78Once a target has been ranged, you may also use themil dot reticle to quickly estimate proper hold-over on longer shots. In order to do this, you will have to be very familiar with the ballistics of your particular weapon and ammunition at all distances. It can be very helpful to keep a printed ballistic chart handy. As always, your scope must be set to 14X magnification.For example, lets say you’ve ranged a deer with your mil dot and determined that he is 300 yards away. You’ve zeroed in your rifle at 100 yards, and know through practice and ballistics info that your bullet will drop 11” at 300 yards. You know that the mil spacing on the reticle is equal to 3.6” at 100 yards. This means the mil spacing will be 10.8” at 300 yards. Therefore, to make your shot you’ll need to hold the center crosshair about 1 mil high from the deers vital zone.Holdover:910Vortex Optics ® believes strongly in responsible, ethical hunting and a word should be said about the difficulty of long range shooting at game. Although reticles like the Vortex “Mil Dot” can make long distance shots much easier, there are still many variables at play affecting every shot. This type of shooting is not easy – plenty of practice is essential. Everyone doing this kind of shooting should also learn their personal effective range, and NOT shoot beyond it at game. Y our effective range will depend on what you’re shooting at: for big game, it might be the range at which you can keep all your shots inside of ten inches, for smaller game it might be the range that all your shots can be kept inside of three inches.Be responsible. The keys are knowing your rifle,ammunition and your own abilities!Unlimited • Unconditional • TransferableWe at Vortex Optics want you to use and enjoy your optics with complete confidence . . . that’s why our V.I.P. warranty is so straightforward. Should your Vortex Optics product ever require service, we will repair or replace it absolutely free - no matter what the cause!The VIP warranty is a Very Important Promise to you . . . because you are a very important person to us. Each Vortex binocular, spotting scope and riflescope is built to last, and unconditionally guaranteed with our commitment to your absolute satisfaction.© Vortex Optics USAVortex Optics2120 West Greenview Drive Middleton, WI 53562(800)******************************。
Drexel University,
Christopher D.CeraIlya Braude Geometric and Intelligent Computing Laboratory,Department of Computer Science,Drexel University,Philadelphia,PA19104Taeseong KimMobile Multimedia Laboratory, LG Electronics Institute of Technology,Seoul,137-724,KoreaJungHyun Han Department of Computer Science andEngineering,Korea University,Seoul,136-701,KoreaWilliam C.Regli Geometric and Intelligent Computing Laboratory,Department of Computer Science,Drexel University,Philadelphia,PA19104Hierarchical Role-Based Viewing for Multilevel Information Security in Collaborative CAD Information security and assurance are new frontiers for collaborative design.In this context,information assurance(IA)refers to methodologies to protect engineering infor-mation by ensuring its availability,confidentiality,integrity,nonrepudiation,authentica-tion,access control,etc.In collaborative design,IA techniques are needed to protect intellectual property,establish security privileges and create“need to know”protections on critical features.This paper provides a framework for information assurance within collaborative design based on a technique we call Role-Based Viewing.We extend upon prior work to present Hierarchical Role-Based Viewing as a moreflexible and practical approach since role hierarchies naturally reflect an organization’s lines of authority and responsibility.We establish a direct correspondence between multilevel security and mul-tiresolution surfaces where a hierarchy is represented as a weighted directed acyclic graph.The permission discovery process is formalized as a graph reachability problem and the path-cost can be used as input to a multiresolution function.By incorporating security with collaborative design,the costs and risks incurred by multiorganizational collaboration can be reduced.The authors believe that this work is thefirst of its kind to unite multilevel security and information clouded with geometric data,including multi-resolution surfaces,in thefields of computer-aided design and collaborative engineering.͓DOI:10.1115/1.2161226͔1IntroductionInformation assurance͑IA͒refers to methodologies to protect and defend information and information systems by ensuring their availability,integrity,authentication,confidentiality,and nonrepu-diation.In collaborative design,IA is mission-critical.Suppose a team of designers is working collaboratively on a3D assembly model.Each designer has a different set of security privileges and no one in the team has the“need to know”the details of the entire design.In collaboration,designers must interface with others’components/assemblies,but do so in a way that provides each designer with only the level of information he or she is permitted to have about each of the components.For example,one may need to know the exact shape of some portion of the part͑includ-ing mating features͒being created by another designer,but not the specifics of any other aspects of the part.Such a need can also be found when manufacturers outsource designing a subsystem: manufacturers may want to hide some critical information of the entire system from suppliers.The authors believe that a geometric approach to IA represents a new problem that needs to be addressed in the development of collaborative CAD systems.The approach we develop has many uses visible across several significant scenarios we envision for applying this work:Protection of sensitive design information:As noted above, designers may have“need to know”rights based on legal,intel-lectual property,or national security requirements. Collaborative supply chains:Engineering enterprises out-source a considerable amount of design and manufacturing activ-ity.In many situations,the organization needs to provide vital design data to one partner while protecting the intellectual prop-erty of another partner.Multidisciplinary design:For designers of different disci-plines working on common design models,designers suffer from cognitive distraction when they must interact with unnecessary design details that they do not understand and cannot change.For example,an aircraft wheel well͓1͔is a complex and confusing place in which electronics,mechanical,and hydraulics engineers all must interact in close quarters with vast amounts of detailed design data.These interactions could be made more efficient if the design space could be simplified to show each engineer only the details they need to see.This paper develops a new technique for Role-Based Viewing ͓2͔in a collaborative3D assembly design environment,where multiple users work simultaneously over the network,and pre-sents a combination of multiresolution geometry and multilevel information security models.Among various issues in IA,access-control is critical for the purpose.We demonstrate the specifica-tion of access privileges to geometric partitions in3D assembly models defined based on the Bell-La Padula model.Aside from digital3D watermarking,research on how to provide IA to dis-tributed engineering teams,working in collaborative graphical en-vironments,remains a novel and relatively unexplored area.We achieve these functional capabilities within a system designed for secure,real-time collaborative viewing of3D models by multiple users working synchronously over the internet on standard graph-ics workstations.The contributions of this work,developed in͓2͔, include:Providing a geometric approach to Information Assurance:Our work augments currently practiced access-control techniques in collaborative CAD and PDM systems.Although most of these systems offer access-control facilities,they are often limited to prohibiting access to models and documents and not partitions of geometry.Developing alternatives to the problem of“all-or-nothing”per-Contributed by the Engineering Simulation and Visualization of ASME for pub-lication in the J OURNAL OF C OMPUTING AND I NFORMATION S CIENCE IN E NGINEERING.Manuscript received January26,2004;final manuscript received March24,2005.Assoc.Editor:N.Patrikalakis.2/Vol.6,MARCH2006Copyright©2006by ASME Transactions of the ASMEmissions:The standard method for handling a lack of appropriate permissions is suppression of the sensitive features.This work attempts to highlight some alternatives other than the traditional solution.In our revised method,we show how geometric partitions are used to create multiple level-of-detail͑LOD͒meshes across parts and subassemblies to provide a Role-based View suitable for a user with a given level of security clearance.The specific contri-butions of this paper include:Introducing role hierarchies:We revisit the problem of role-based viewing in an updated context using role hierarchies.A hierarchy is represented as a weighted directed acyclic graph ͑DAG͒,and the permission discovery process is formalized as agraph reachability problem.Outlining the relation between multilevel security͑MLS͒hier-archies and multiresolution surfaces:In͓2͔we introduced a mul-tiresolution envelope to degrade surface features for a sensitive part or subassembly where details need to be withheld.We incor-porate this approach into role-hierarchies where the path-cost is used as input to a multiresolution function.The authors believe that this work represents a unique application of multiresolution surfaces to multilevel information security in computer-aided de-sign and collaborative engineering.This paper is organized as follows:Sec.2describes related work from information assurance,collaborative design,and com-puter graphics communities.Section3first reviews the specifica-tion of security features in thefields of solid modeling and engi-neering as outlined in Ref.͓2͔,then presents Hierarchical Role-Based Viewing.Section4explains the details of our multiresolution security model and outlines its relation to the Role Hierarchy.Section5describes the implementation of our proto-type system,and demonstrates a sample scenario using our stly,Sec.7summarizes our results,presents our con-clusions,and outlines goals for future research.2Related WorkThe contributions presented in this paper are related to infor-mation assurance,collaborative design,and multiresolution sur-face generation.2.1Information Assurance and Security.Current research on information assurance incorporates a broad range of areas fo-cused on protecting information and information systems by en-suring their availability,integrity,confidentiality,nonrepudiation, authentication,and controlling modes of rmation as-surance research,in the context of the CAD domain,has been partially addressed by the computer graphics community through the development of3D digital watermarking͓3͔.Digital Water-marking is used to ensure that the integrity of a model has been maintained,as well as provide a foundation for proof of copyright infringement.Other areas of research have been in authentication and access-control.We will introduce past and present research on access control methodologies and outline the differences between the varying policies.There is a clear distinction between authentication and access control services.Authentication services are used to correctly de-termine the identity of a user.Access control is the process of limiting access to resources of a system only to authorized users, programs,processes,or other systems.Authentication is closely coupled with access control,where access control assumes that users of an information system have properly been identified by the system.If the authentication mechanism of a system has been compromised,then the access control mechanism that follows will certainly be compromised.The primary focus of our work is to articulate an access control policy,specifically for the geometry of a solid model,assuming a robust authentication mechanism has already been established.Access-control literature describes high-level policies on how accesses are controlled,as well as low-level mechanisms that implement those policies.The common access control policies found in literature are Dis-cretionary,Lattice-Based,and Mandatory Access Control͑DAC,LBAC,and MAC,respectively͒.DAC was formally introducedby Lampson͓4͔,where essentially the owner of an object hasdiscretion over what users were authorized to access that object.Access broadly refers to a particular mode of operation such asread or write.The owner is typically designated as the creator ofan object,hence it is an actual user of the system.This is differentfrom LBAC and MAC,which we will refer to collectively asMAC͓5͔,where individual users have no discretion over objectaccess.MAC͓6͔is primarily concerned with theflow of informa-tion,thereby enforcing restrictions on the direction of communi-cation channels.For further discussion on access control policies,we refer interested readers to a survey by Sandhu͓7͔.Role-Based Access Control͑Group policies found in currentoperating systems could be viewed as an instance of hierarchicalRBAC͒.RBAC is an emerging area of study,and is actively pur-sued as an augmentation of traditional DAC and MAC.RBAC isa type of Multi-Level Security͑MLS͒framework,which is anactively pursued area in the database community͓8,9͔.In RBAC,individual users are assigned to roles,and the access permissionsof an object are also assigned to roles.Therefore the permissionsassigned to a role are acquired by the members associated with it.This additional layer reduces the management of permissions andsupports the concepts of least privilege,separation of duties,anddata abstraction.RBAC,and its associated components,are aninstrument for expressing a policy,and not a policy by itself.Forrole-based viewing,we use a MAC policy embodied within an RBAC framework.2.2Collaborative Design.There is a vast body of work on concurrent engineering and distributed collaborative design.Visu-alization and multiuser protocols were the primary focus of manyearly systems,and generally fell under the terms Distributedand/or Collaborative Virtual Environments͓10–17͔.Thereafter,lightweight design͓18–21͔and assembly modeling systems͓22͔for distributed CAD began to emerge.With the demand for distributed CAD and more edit-drivenmodeling mechanisms,several research systems and commercialproducts have reached fruition.Current work in distributed col-laborative design,with respect to geometry,can be“loosely”grouped into two categories:visualization and annotation of CADmodels;co-design and manipulation of CAD geometry.The cur-rent demonstration of our work is primarily targeted at the formercategory.This is complementary to facet-based editing techniques ͓23͔,however editing multiresolution geometry hierarchies can be extended to feature,assembly,and native-geometry editing facili-ties.This assumes a system with policies that permit editing on alow resolution model,where the user in question does not havepermissions to view the full resolution model.Polyinsantiationissues are beyond the scope of this paper,although most theoret-ical issues have been developed with regard to databases͓8,9͔.The generation of facets is inevitable in the visualization pro-cess,and most distributed CAD systems deal with this problem invery diverse ways.System designers must resolve varying trade-offs in the areas of visualization,editing mechanisms,bandwidthutilization,and accuracy.Some systems have targeted view-dependent techniques as an attempt to reduce aggregate band-width and computation͓24–27͔.Current literature now providesnumerous mechanisms and interfaces from which to edit CADmodels:features͓28,29͔;assemblies͓30,31͔;native surfaces͑e.g.,b-rep͓͒32,33͔;and facet-based techniques͓23͔.The heterogeneity of existing systems and interchange formats further complicates the domain.Reference͓28͔contains an extensive overview of current distributed CAD systems from the perspective of architec-ture and data exchange.2.3Multiresolution Techniques.Polygon meshes lend them-selves to fast rendering algorithms,which are hardware-accelerated in most platforms.Most CAD models are representedJournal of Computing and Information Science in Engineering MARCH2006,Vol.6/3as a set of trimmed parametric surfaces,so tessellation to a desired resolution is performed ͓34,35͔.Many applications,including CAD,require highly detailed models to maintain a convincing level of realism.It is often necessary to provide LOD techniques in order to deliver real-time computer graphics and animations.Therefore,mesh simplification is adopted for efficient rendering and transmission.The most common use of mesh simplification is to generate multiresolution models or various levels of detail ͑LOD ͒.For example,closer objects are rendered with a higher LOD,and distant objects with a lower LOD.Thanks to LOD management,many applications such as CAD visualization can accelerate rendering and increase interactivity.A survey on mesh simplification can be found in Ref.͓36͔.The most popular polygon-reduction technique is an edge col-lapse or simply ecol ͑more generally,vertex merging or vertex pair contraction ͒where two vertices are collapsed into a single one.The issues in ecol include which vertices to merge in what order,where to place the resulting vertex,etc.Vertex split or sim-ply vsplit is the inverse operation of ecol .These operations are illustrated in Fig.1͑a ͒and a sequence of operations is illustrated on a sample model given in Fig.1͑b ͒.Hoppe proposed the progressive mesh ͑PM ͓͒38͔,which con-sists of a coarse base mesh ͑created by a sequence of ecol opera-tions ͒and a sequence of vsplit operations.Applying a subset of vsplit operations to the base mesh creates an intermediate simpli-fication.The vsplit and ecol operations are known to be fast enough to apply at runtime,therefore supporting dynamic simpli-fication.3Role-Based ViewingIn the context of 3D design,a model M is a description of an artifact,usually an individual part or assembly,in the form of a solid model.A true collaborative engineering environment enables multiple engineers to simultaneously work with M .The engineers ͑designers,process engineers,etc ͒correspond to a set of actors A =͕a 0,a 1,...,a n ͖each of which has associated with it a set of roles .Roles,R =͕r 0,r 1,...,r m ͖,define access and interaction rights for the actors.For example,actor a 3might have associated with it roles,r 20,r 23,and r 75—this entitles a 3to view ͑and per-haps change ͒portions of M associated with these roles.Portions of M not associated with these roles,however,might be “off lim-its”to actor a 3.This section will build on the results of Ref.͓2͔,where Role-based Viewing was developed in the context of dis-tributed collaborative CAD,by introducing role hierarchies and their relation to multiresolution surfaces.We formulate the problem of role-based viewing in the follow-ing subsections by developing:͑a ͒Actor-role framework:a general RBAC framework for describing actors and roles within a collaborative-distributed design environment.This framework uses a hierarchical graph to capture role-role relationships and create a relation between actors and roles.͑b ͒Model-role framework:an associative mapping from roles to topological regions on models.These regionscapture the security features ,F ,of a 3D model—relating how a point,patch,part,or assembly can be viewed by actors with given roles.͑c ͒Hierarchical role-based viewing:an algorithm to generate a role-based view given an actor a ,his/her set of roles,the role hierarchy ͑RH ͒,a model M ,and its set of secu-rity features.A role-based view is a tailored 3D model which is customized for actor a based on the roles defin-ing a ’s access permissions on the model.In this way,the role-based view model does not compromise sensitive model information which a is not allowed to see ͑or see in detail ͒.This is accomplished using a mesh simplifica-tion technique to generate the role-based view .3.1Actor-Role Security Framework.Our security frame-work is based on an adaptation of role-based access control,as developed in the information assurance and security literature ͓39͔,to the collaborative design problem.We focus on the relation between actors,their roles and the solid model geometry.This is in contrast to other work on access control in collaborative CAD which has focused mainly on database synchronization/transaction issues ͓40͔.Representing actors and roles:We define a hierarchical RBAC framework where:͑1͒Entities include a set of actors,A =͕a 0,a 1,...,a n ͖and a setof roles r =͕r 0,r 1,...,r m ͖;͑2͒Actor-role assignment,AR,is a relation ͑possibly many-to-many ͒of actors to roles:AR ʕA ϫR ;͑3͒Role hierarchy,RH,captures the relationships among theroles.For example,the permissions entailed by role r 75might be a superset of those entailed by.Hence,the role hierarchy is a weighted ,directed acyclic graph ͑DAG ͒,RH=͑R ,H ͒,where H ʚR ϫR is the hierarchical set of re-lationships ͑edges ͒among the roles in R .This creates a partial order on R ,hence ͑in the example above ͒if ͗r 23,r 75͘H then r 23՞r 75.The weight of each edge in H is given by the real-valued function w :H →͓0,1͔.An example of this RBAC framework is given in Fig.2.For the remainder of this paper,we focus on read/viewing permissions granted by a given set of roles.Rather than “all or nothing”read permissions,our objective is to assign a “degree of visibility”to features of a model based on an actor’s ing this formu-lation,we show how one can implement a Bell-La Padula-based ͓6͔security model for collaborative viewing of CADdata.Fig.1Illustration of multiresolution techniques.…a …Edge collapse …ecol …and vertex split …vs-plit …operations.v t …top …and v b …bottom …collapse into v m …middle ….The inverse operation in-volves splitting v m back into v t and v b .…b …Sequence of operations on the “socket”model †37‡.Fig.2Example actor-role …AR …and role hierarchy …RH …assignments4/Vol.6,MARCH 2006Transactions of the ASMEExample :Using the simple actor-role assignment matrix and role hierarchy from Fig.2,we can compute the degree of visibility to each actor for a model assigned to a specific role.To implement the Bell-La Padula ͓6͔model,we need to compute visibility in such a way as to guarantee that the role ͑e.g.,security clearance ͒of someone receiving a piece of information must be at least as high as the role assigned to the information itself.In this way,a CAD model classified as “Secret”can only be viewed by those with a “Top-Secret”or “Secret”classification,but not viewed by someone with only a “Confidential”level of access.Figure 3il-lustrates this example.3.2Model-Role Security Framework.Let M be a solid model of an artifact ͑part,assembly,etc.͒and let b ͑M ͒represent the boundary of M .In this context,the Model-Role Assignment ,MR,is a relation ͑possibly many-to-many ͒assigning roles to points on the surface of the model:MR ʕb ͑M ͒ϫR ,where each point on b ͑M ͒has at least one role ͑i.e.,"p b ͑M ͒,$r R such that ͗p ,r ͘MR ͒.In this way,each point on the surface of the solid model M has associated with it some set of access rights dependent on the roles associated with it.In practice,it is impractical to assign roles point-by-point to the b ͑M ͒.Hence,we define a set of security features ,f =͕f 0,f 1,...,f k ͖,where each f i is a topologically connected patch on b ͑M ͒and ഫF =b ͑M ͒;and each f i has a common set of role assignments.Therefore,the Model-Role assignment can be sim-plified to be the relation associating security features with access roles:MR ʕF ϫR .Example :Let M be a solid model;let,F =͕f 1͖,where f 1=b ͑M ͒͑i.e.,the entire boundary is one security feature ͒.If MR =͕͗f 1,r 0͖͑͘where r 0is from the previous example in Fig.3͒,then we can see the resultant model for r 0depicted in Fig.4.3.3Hierarchical Role-Based Viewing.The issue now is that,for a given actor a ,what portions of the model M that he/she can see will depend on their associated roles and the security features of the model.Depending on their permissions,a new model,M Ј,must be generated from M such that the security features are not shown or obfuscated based on the actor’s roles.If their roles give them permission to see certain features ͑i.e.,mating features ͒,then the resulting model includes the features with the same fidelity ͓41͔as in M ;if not,the features must be obfuscated in such a way as to hide from a what a does not have the right to see.Hence,the role-based view generation problem can be stated:Problem :Given a set of roles and their relationships ͑R and RH ͒;a solid model and its security features ͑M ,F ,and MR ͒;and an actor ͑a and AR ͒,determine the appropriate view M Јof modelM for actor a .We propose a solution based on the use of multiresolution meshes,as follows:͑1͒Convert solid model M to a high-fidelity mesh representa-tion;͑2͒Based on F ,determine which facets belong to each securityfeature,f ;͑3͒For each security feature f ,do:͑a ͒If the intersection of actor a ’s roles and f ’s roles isnonempty,then add the facets associated with f to M Ј;͑b ͒If actor a ’s roles do not intersect the roles of f ,deter-mine ͑using RH ͒how much of f they are allowed to see and create a set of modified facets to represent f for inclusion in M Ј.͑4͒Clean up the resulting M Јso that boundaries of the f i ’s aretopologically valid.͑5͒Return M Ј.There are three research problems we address:͑1͒How does the role-hierarchy RH relate to the degree ofvisibility?We show how the weighted DAG that comprises RH can be used to implement a number of useful security policies by making the model quality a function of the path-cost among roles in RH.͑2͒How to modify the facets for each f i based on RH?Our approach is to use a security policy ͑based on Bell-La Padula ͒associated with the role hierarchy RH to determine how to modify the model.In some cases,policy will dictate degradation of the model fidelity;in other cases,the security features may be completely deleted or replaced with a simple convex hull or bounding box as in Ref.͓2͔.To accomplish this,we employ multiresolution meshes:model fidelity will be preserved to the degree the actor’s rights allow it.The result is a mesh appropriate for viewing by the actor a .͑3͒How to ensure that the resulting regions form a topologi-cally valid model?Deforming the model feature by feature may result in topological regions of facets in M Јthat are misaligned or aesthetically unpleasing.Cracks and occlusion can be avoided by preserving the boundary edges during simplification.Example :This example shows a model M whose surface is described by one security feature f 0.Given the role-hierarchy from Fig.3,and four actors,a 0,a 1,a 2,and a 3with their AR shown in Fig.2.Figure 5shows the four different views of model M they each see.Given the AR,RH,and MR assignments,we can derive the direct actor ϫfeature mappings.Figure 6gives the di-rect mappings specified implicitly by the AR,RH,and MR given in Figs.2͑a ͒,2͑b ͒,and 5respectively.The two MR assignments that are not shown are f 1r 1and f 2r 2.It is important to note that,similar to inheritance found in most object-oriented program-ming languages,a 0cannot see f 1or f 2even though it is the base role for subroles r 1,r 2,and r 3.Hence an inheritance relation al-lows a child to inherit the permissions of the parent,but nothing is implied in the other direction.4Technical ApproachWe combine techniques from solid modeling and computer graphics to provide a secure collaborative environment which sup-ports real-time design.In this section we describe how to modify and configure Hierarchical RBAC to support our multiresolution security model.We describe the problems,algorithms employed,and finalconsiderations.Fig.3An example weighted role hierarchy with associatedlabelsFig.4An example part with one security feature where b …M …is assigned to r 0Journal of Computing and Information Science in EngineeringMARCH 2006,Vol.6/54.1Hierarchical RBAC Policy.Since RBAC is a means of articulating policy rather than a policy by itself,an actual policy is necessary.We wish to adopt a policy similar to the classical MAC model ͓6͔.This is defined in terms of the following axioms using to return the security level of either an actor or a feature:͑1͒Simple security property:Actor a can read feature f iff͑a ͒ജ͑f ͒.This is also known as the read-down property.͑2͒Liberal ء-property:Actor a can write feature f iff ͑a ͒ഛ͑f ͒.This is also known as the write-up property.There are many variations of the ء-property,but we will focus on the simple security property which essentially states that the clearance of a person receiving a piece of information must be at least as high as the classification of the object.Details on a formal construction of MAC in RBAC have been presented by Osborn ͓5͔.Hierarchical RBAC is a natural means for structuring roles thatreflect an organization’s lines of authority and responsibility ͓39͔.The main distinction between our approach and the generic RBAC frameworks found in literature,is that we also allow per-missions to be modified through the role hierarchy .Typically per-missions ͑i.e.,an object and a permissible operation ͒are associ-ated with every combination of object ϫrole .Since our read permissions are specified by a degree of visibility value,an inher-itance relation can further refine this value.An inheritance relation is a binary relation ͑parent ,child ͒,where the child inherits per-missions from the parent based upon a multiplicative weight w .For instance:w =1.0preserves the parents permissions exactly,while w =0.5will reduce the degree of visibility by half for all inherited objects.By transitivity,this weighted factor applies to all inherited objects specified in the role hierarchy.Intuitively,it might appear that we are breaking the simple se-curity property by allowing some actors to view objects that they normally would not be able to see.This is not the case,and in-stead should be viewed as transforming one object into a new object that is permissible.Hence,our model still adheres to the simple security property.Given an actor ͑a ͒and a feature ͑f ͒,the test to determine if a has permissions on f is equivalent to computing graph reachability among all possible pairs of roles assigned to both a and f .We will use R a to denote the set of roles assigned to a ,and R f for the set of roles assigned to f .If any role in R a is reachable from any other role in R f ͑i.e.,there exists a path ͒,then the product of all weights along the path yields the degree of visibility for that path.We will use a reachability function to return the set of all roles reachable from a given role.Several paths are possible,hence the resultant degree of visibility for a will be chosen as the maximum.We denote the function that returns the maximum degree of visibility for a on f as ␣͑a ,f ͒.The result of this function can be computed once,stored,and reused until an existing role assignment ͑AR or MR ͒is modified.The degree of visibility is then used as a param-eter to another function which degrades the fidelity of a feature depending on an actors permissions.4.2Generation of Multilevel Security Models.For part/component/assemblies with regions that need to be secured,mul-tiresolution techniques are employed to provide various levels of detail.Although the original ͑highest ͒resolution version of a model might be a breach for some actors,lower resolution LODs will be sufficiently secure to transmit to those actors.In addition to purely geometric multiresolution techniques,Shyamsundar and Gadh have developed a framework for representing different lev-els of detail for geometric feature data ͓31,42͔.Our security model could be used in conjunction with this feature LOD representa-tion,but an automatic simplification algorithm needs to be developed.Mesh simplification techniques include either vertex decima-tion,vertex clustering,or edge contraction.Choosing a specific simplification technique among the breadth of candidates is appli-cation dependent.To address the demands of an interactive col-laborative design environment,we outline several issues which are critical for simplification:͑1͒Speed :As the number of component/assemblies in a sessionincreases,the simplification becomes the bottleneck.We need an algorithm capable of drastic simplification in the least amount of time.͑2͒Continuous :A continuous spectrum of detail is necessaryso an appropriate model can be selected at runtime.We do not wish to store all possible LODs within the model re-pository due to space constraints,but parameters of the simplification hierarchy can be precomputed and stored.͑3͒Boundary preserving :The boundary of objects should bepreserved in order to distinguish objects from one another.Inadvertent occlusion and cracks may result if we relieve this constraint.͑4͒View-independence :The viewer receives 3D modelinfor-Fig.5An example part with one security feature …f 0…consist-ing of b …M …assigned to r 0,a set of actors assigned to roles,and their corresponding set of securemodelsFig.6The direct permission mappings derived from the AR,RH,and MR relations given in Figs.2…a …,2…b …,and 5,respectively6/Vol.6,MARCH 2006Transactions of the ASME。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Translating a Planar Objectto Maximize Point ContainmentPankaj K.Agarwal,Torben Hagerup,Rahul Ray,Micha Sharir,Michiel Smid,and Emo WelzlCenter for Geometric Computing and Department of Computer Science,Duke University,Durham,NC27708-0129,U.S.A.pankaj@Institut f¨u r Informatik,Universit¨a t Frankfurt,D–60054Frankfurt am Main,Germany.hagerup@rmatik.uni-frankfurt.deMax-Planck-Institute for Computer Science,66123Saarbr¨u cken,Germany.rahul@mpi-sb.mpg.deSchool of Computer Science,Tel Aviv University,Tel Aviv69978,Israel;and Courant Institute of Mathematical Sciences,New York University,New York,NY10012,USA.sharir@math.tau.ac.ilSchool of Computer Science,Carleton University,Ottawa,Canada K1S5B6.michiel@scs.carleton.caInstitut f¨u r Theoretische Informatik,ETH Z¨u rich,CH–8092Z¨u rich,Switzerland.emo@inf.ethz.chAbstract.Let be a compact set in and let be a set of points in.Weconsider the problem of computing a translate of that contains the maximumnumber,,of points of.It is known that this problem can be solved in a timethat is roughly quadratic in.We show how random-sampling and bucketingtechniques can be used to develop a near-linear-time Monte Carlo algorithm thatcomputes a placement of containing at least points of,for given ,with high probability.We also present a deterministic algorithm that solves the-approximate version of the optimal-placement problem for convex-gonsin time,for arbitrary constant.1IntroductionLet be a compact set in and let be a set of points in.We define the optimal-placement problem to be computing a point so that the translateof contains the maximum number of points of.SetMotivated by applications in clustering and pattern recognition(see[14,16]),the optimal-placement problem has received much attention over the last two decades. Chazelle and Lee[7]presented an-time algorithm for the case in which isa circular disk,and Eppstein and Erickson[11]proposed an-time algorithmfor rectangles.Efrat et al.[10]developed an algorithm for convex-gons with a run-ning time of,where,which was subsequently improved by Barequet et al.[3]to.All the algorithms above,except the one for rectangles,require at least quadratictime in the worst case,which raises the question of whether a near-linear approxima-tion algorithm exists for the optimal-placement problem.In this paper we answer the question in the affirmative by presenting Monte-Carlo and deterministic approximation algorithms.We call a translate of an-approximate placement if.An algorithm that produces an-approximate placement is called an-approximation algorithm.We define the-optimal-placement problem to be the onethat asks for an-approximate placement.We make the following assumptions about:(A1)The boundary of,denoted by,is connected and consists of edges,whose endpoints are called vertices of.Each edge is described by a vector function ofa scalar parameter,each component of which is a polynomial of bounded degree.We will refer to as a disk.(A2)The boundaries of any two translates of intersect in at most points,and the intersections can be computed in time.By computing an intersection,wehere mean determining the two edges,one for each translate,that are involved inthe intersection,as well as the two corresponding values of the scalar parameter.Moreover,for every point,we can decide in time whether.(A3)is sandwiched between two axes-parallel rectangles whose widths and heights differ by factors of at most and,respectively,for.We call fat if and in assumption(A3)are bounded by constants.Schwarzkopf et al.[19] showed that after a suitable affine transformation,a convex polygon is fat with.We assume a model of computation in which the roots of a bounded-degree poly-nomial can be computed in time,which implies that primitive operations on theedges of,e.g.,computing the intersections between two edges of two translates of,can be performed in time.In Section2,we present two algorithms for the optimal-placement problem andshow how bucketing can be used to expedite the running time,especially if is fat.In particular,let be the running time of an algorithm for the optimal-placement problem on a set of points,and let denote the time required to partitionpoints into the cells of an integer grid.Then the bucketing algorithm can compute an optimal placement in time,where.Besidesbeing interesting in its own right,this will be crucial for the approximation algorithms.In Section3,we show that using random sampling and/or bucketing,we can trans-form any deterministic algorithm for the optimal-placement problem to a Monte-Carlo algorithm for the-optimal-placement problem.Given a parameter,theTime ReferenceSection3.1Section3.2Section3.2;see Table1.If is fat and,by combining two levels of random sampling with the buck-eting technique,we can compute an-approximate placement in time with errorprobability at mostwhere,and sort them pute the depth of one intersection with respect to by brute force and step through the remaining ones in sorted order while maintaining the depth.Finally report a point of the maximum depth encountered for any.The total time spent is.Alternatively,we can compute using the algorithm by Amato et al.[2],tra-verse the type2vertices of,e.g.,in depth-first order,and compute a vertex of maximum depth(with respect to).Since has vertices,this algo-rithm takes time.Hence,we obtain the following. Theorem1.Let be a set of points in the plane and let be a disk satisfying as-sumptions(A1)–(A3).The value of can be computed in timeor in time.If,then,and if is convex,then[15] and[18].(For the upper bounds on and,an -time preprocessing step is needed.)Therefore Theorem1implies the following. Corollary1.Let and be as above.Then can be computed in time if has edges,and in or time if is a convex-gon.Bucketing and estimating.For any two positive real numbers and,we denote by the two-dimensional grid through the origin whose cells have horizontal and vertical sides of lengths and,respectively.Hence,each cell of is of the form for some integers and.We call the pair the index of.We need an algorithm that groups the points of according to the cells of some grid,i.e.,stores in a list such that for each grid cell,all points of in occur together in a contiguous sublist.This operation is similar to a sorting of by grid cell,but does not require the full power of sorting.Let denote the time needed to perform such a grouping of points according to some grid and assume thatis nondecreasing and smooth in the sense that(informally,a smooth function grows polynomially).The following lemma is straightforward. Lemma1.Let be a set of points in and let be a disk satisfying assumptions (A1)–(A3).Let be such that for axes-parallel rectanglesof width and height and of width and height.Let be the maximum number of points of contained in any cell of the grid.Then.Lemma1shows that an approximation to withcan be computed in time.Let us see how the grouping of can be implemented.It is clear that once each point of has been mapped to the index of the cell containing of a grid under consideration,can be grouped with respect to the grid in time by sorting the pairs lexicographically. The mapping of points to grid indices uses the nonalgebraicfloor function.To avoid this,we can replace the grid by the degraded grid introduced in[8,17],which can beconstructed in time without using thefloor function,and for which Lemma1 also holds.Given any point,the cell of the degraded grid that contains can be found in time,so that the grouping can be completed in time.In a more powerful model of computation,after mapping to grid indices,we can carry out the grouping by means of bining the universal class of Dietzfel-binger et al.[9]with a hashing scheme of Bast and Hagerup[4],we obtain a Las Vegas grouping algorithm thatfinishes in time,except with probability at most for somefixed.We omit the details.A bucketing algorithm.We can use Lemma1and Theorem1to obtain a faster algorithm for computing in some cases.Suppose we have an algorithm that computes in ing Lemma1,wefirst compute and,for each pair,consider the grid obtained by shifting by the vector .For each cell of that contains at least points of,we run on the set to obtain a point of maximum depth with respect to.Finally we return a point for which is maximum over all runs of.To see the correctness of the algorithm,let have.Observe that for some,lies in the middle ninth of some cell of,in which case .It is now clear that and.Let us analyze the running time.For each of,the algorithm spends time to partition among the grid cells.Since at most cells of contain at least points of and no cell contains more than points,the total running time is,where in the last step we used the relation and the assumption that is nondecreasing. We thus obtain the following result.Theorem2.Let be a set satisfying assumptions(A1)–(A3),let be a set of points in,and let be an algorithm that computes in time.Then can be computed in time.Corollary2.Let and be as above.The value of can be computed in time if is fat and has edges,and inor time if is a convex-gon. 3Monte-Carlo AlgorithmsIn this section we present Monte-Carlo algorithms for the-optimal-placement prob-lem.These algorithms use one of the deterministic algorithms described in Theorem1 as a subroutine.We will refer to this algorithm as and to its running time as. We assume that and are nondecreasing and that is smooth.3.1A random-sampling approachWefirst present an algorithm based on the random-sampling technique.We carry out the probabilistic analysis using the following variant of the well-known Chernoff bounds (see,e.g.,Hagerup and R¨u b[13]).Lemma2.Let be a binomially distributed random variable and let.1.For every,.2.For every,.Theorem3.For arbitrary,an-approximate solution to the optimal-placementproblem can be computed in time with error probability at most,where.Proof.Let and.The algorithmfirst draws a-sample of,i.e.,includes every point of in with probability and independently of all other points.If(the sampling fails),the algorithm returns an arbitrary point.Otherwise it uses to return a point of maximum depth with respect to.Since and is smooth,it is clear that the algorithm can be executed in time.By Lemma2,the sampling fails with probability at most. If or,the output is obviously correct.Assume that this is not the case and that the sampling succeeds.Let us write for and for and let be a point with. Informally,our proof proceeds as follows.Let be the set of“bad”points.The error probability is equal to.Wefirst show that is likely to be large,where“large”means at least.Subsequently we show that for every,is not likely to be bining the two assertions shows that except with small probability,.Thefirst part is easy:Since and,Lemma2implies thatNowfix.Since and,we haveThe preceding argument applies to afixed.A priori,we have to deal with an infinite number of candidate points.However,using the fact that the arrangement defined by the boundaries of the sets,with,has vertices of type2, it is not difficult to see that there is a set with such that for every ,there is a with.Therefore the probability thatfor some is.The other failure probabilities identified above are no larger.Now,,,and.Moreover, we can assume that,since otherwise the theorem claims nothing.But then.Therefore,except if is bounded by some constant(in which case we can use a deterministic algorithm),the failure probability is at most.Combining Theorem3with Corollary1,we obtain the following result.Corollary3.For a given set of points,a set and arbitrary,an-approximate placement can be computed with error probability at most in time if has edges,and in ortime if is a convex-gon.Our random-sampling approach can also be used to solve related problems.As anexample,consider the problem of computing a point of maximum depth in a set of halfplanes.If we denote this maximum depth by,then.By computing and traversing the arrangement defined by the bounding lines of the halfplanes,onecan compute in time.Since a corresponding decision problem is3SUM-hard (see Gajentaan and Overmars[12]),it is unlikely that it can be solved in subquadratic time.If we apply our random-sampling transformation with.3.2Bucketing and sampling combinedWe now present a Monte Carlo algorithm that combines Theorem3with the bucketing algorithm described in Section2.First compute,as defined in Lemma1.Next,for each pair, consider the grid as in Section2.Fix a parameter.For each cell of with,run the algorithm described in Section3.1on the set to obtain a point and compute the value.Finally return a point for which is maximum.Theorem5.Let be a disk satisfying assumptions(A1)–(A3).For arbitrary, an-approximate placement of can be computed intime by a Monte Carlo algorithm with error probability at most. Corollary4.For arbitrary,an-approximate placement of can be computed in time with probability of error at most, if is a convex-gon.Proof.We prove thefirst claim.Let be the algorithm of Corollary1.Then .Applying Theorem5to,we obtain an-approximation algorithm for the optimal-placement problem with running time and error probability .If we choose. Proof.We use the following algorithm,which might be described as sampling fol-lowed by bucketing followed by sampling:Draw a random-sample of,wherefor a constant to be chosen below.If,return an arbitrary point.Otherwise apply the algorithm of Corollary4to,but with approx-imation parameter,rather than,and return the point returned by that algorithm. The overall running time is clearly.As in the proof of Theorem3,we write for and for.Assume thatand that and let be the maximum value of over all points in the plane.By Lemma2,.Moreover,the analysis in the proof of Theorem3shows the probability that for somewith to be.Finally,the probability that the algorithm of Corollary4returns a point with is at most.If ,the probability under consideration is.4A Deterministic Approximation AlgorithmIn this section we present a deterministic approximation algorithm that solves the optimal-placement problem.For simplicity,we assume that is convex and has edges. The algorithm can be extended to the general case assumed above.For example,at the cost of an-factor in the running time,the algorithm can be extended to arbi-trary convex-gons.As above,let be the set of translates,with,and let denote the arrangement defined by the boundaries of the elements of.4.1CuttingsWe will refer to a simply connected region with at most four edges—left and right edges being vertical segments and top and bottom edges being portions of the bound-aries of respective translates of—as a pseudo-trapezoid.For technical reasons, we will also regard vertical segments and portions of the boundaries of translates of as-dimensional pseudo-trapezoids.For a pseudo-trapezoid and a set of translates of,we will use to denote the set of all elements of whose boundaries intersect.The vertical decomposition of partitions the plane into pseudo-trapezoids.Let be a set of translates of in the plane and let be a pseudo-trapezoid.We will denote by the number of pairs of elements of whose boundaries intersect inside.If is the entire plane,we use the notation to denote,for brevity.Given a parameter,a partition of into a collection of pairwise openly-disjoint pseudo-trapezoids is called a-cutting ofif for every pseudo-trapezoid.The conflict list of a pseudo-trapezoid in is the set.We state the following technical result in full generality,since it is of independent interest,and mayfind additional applications.We apply it here only with being the whole plane.Theorem6.Let be a set of translates of,let be a pseudo-trapezoid,let ,and let be an arbitrarily small constant.A-cutting of of size,along with the conflict lists of its pseudo-trapezoids,can becomputed in time,where the constants of proportionality depend on.We prove this theorem by adapting Chazelle’s cutting algorithm[6]to our setting. We call a subset of a-approximation if,for every pseudo-trapezoid,Next,we call a subset of a-net of if,for any pseudo-trapezoid, implies that.A-net is called sparse for ifAs in[6],we can prove the following.Lemma4.Given a set of translates of,a pseudo-trapezoid,and a parameter ,we can compute,in time,a-net of size that is sparse for .Proof of ing Lemmas3and4,we compute a-cutting ofas follows.Let be a sufficiently large constant whose value will be chosen later.For every,we compute a-cutting of.Thefinal cutting is a-cutting of.While computing,we also compute the conflict lists of the pseudo-trapezoids in.is itself,and is the conflict list of.We compute from as fol-lows.For each pseudo-trapezoid,if,then we do nothing in .Otherwise,wefirst compute a-approximation of of sizeand then a-net of of size that is sparse for.Note thatis a-net of.We then compute the vertical decomposition of within.consists of cells.We replace with the pseudo-trapezoids of.Repeating this for all,we form from.It is easy to see that is a-cutting of.By an analysis similar to the one in[6],it can be shown that the size of thefi-nal cutting is and that the running time of the algorithm is.This completes the proof of Theorem6.By a result of Sharir[20],,where,so Theorem6 implies the following.Corollary6.Let be a set of translates of,let,and let be an arbitrarily small constant.A-cutting of size,along with the conflict lists,can be computed in time.4.2The approximation algorithmLet be real numbers.We now present a deterministic algorithm for computing an integer such that,where.We will need the following lemma.Lemma5.Let be a set of translates of,and let.Given a-cutting of,let denote the maximum depth of any vertex of any pseudo-trapezoid in. Then.Moreover,can be computed in time. Proof.Let be a pseudo-trapezoid of.The maximum depth of any point inside is at most plus the depth of any vertex of.It thus suffices to compute the depth of every vertex of and to return the maximum value among them.We can compute the depths of all the vertices of by following an Eulerian tour on the dual graph of the planar subdivision induced by;see,e.g.,[1].As shown in[1],the time spent in this step is proportional to the total size of all the conflict lists in.Since the size of each conflict list is at most,the claim follows.Our algorithm works in two stages.In thefirst stage,we estimate the value of to within a factor of9,and then we use Lemma5to compute an-approximation of.ing the bucketing algorithm of Section2,we obtain a coarse estimate ofthat satisfies.(Since we assume that is convex,we have ,as follows from[19],which leads to the constant9in the above estimate.) II.We setAs shown in Section2,Step I can be implemented in ing The-orem6and Lemma5,Step II takes timeHence,we conclude the following.Theorem7.Let and be as above,and let be a real number.Intime,for any constant,we can compute an integer with, where.References1.P.K.Agarwal,B.Aronov,M.Sharir,and S.Suri.Selecting distances in the plane.Algorith-mica,9:495–514,1993.2.N.M.Amato,M.T.Goodrich,and puting the arrangement of curvesegments:Divide-and-conquer algorithms via sampling.In Proc.11th ACM-SIAM Sympos.Discrete Algorithms,pages705–706,2000.3.G.Barequet,M.Dickerson,and P.Pau.Translating a convex polygon to contain a maximumnumber of put.Geom.Theory Appl.,8:167–179,1997.4.H.Bast and T.Hagerup.Fast and reliable parallel hashing.In Proc.3rd Annual ACMSymposium on Parallel Algorithms and Architectures,pages50–61,1991.5.H.Br¨o nnimann,B.Chazelle,and J.Matouˇs ek.Product range spaces,sensitive sampling,and derandomization.SIAM put.,28:1552–1575,1999.6. B.Chazelle.Cutting hyperplanes for divide-and-conquer.Discrete Comput.Geom.,9:145–158,1993.7. B.Chazelle and D.T.Lee.On a circle placement puting,36:1–16,1986.8. A.Datta,H.-P.Lenhof,C.Schwarz,and M.Smid.Static and dynamic algorithms for-pointclustering problems.J.Algorithms,19:474–503,1995.9.M.Dietzfelbinger,T.Hagerup,J.Katajainen,and M.Penttonen.A reliable randomizedalgorithm for the closest-pair problem.J.Algorithms,25:19–51,1997.10. A.Efrat,M.Sharir,and puting the smallest-enclosing circle and related prob-put.Geom.Theory Appl.,4:119–136,1994.11. D.Eppstein and J.Erickson.Iterated nearest neighbors andfinding minimal polytopes.Discrete Comput.Geom.,11:321–350,1994.12. A.Gajentaan and M.H.Overmars.On a class of problems in computational geome-put.Geom.Theory Appl.,5:165–185,1995.13.T.Hagerup and C.R¨u b.A guided tour of Chernoff rm.Process.Lett.,33:305–308,1990.14. D.P.Huttenlocher and S.Ullman.Object recognition using alignment.In Proc.1st Internat.put.Vision,pages102–111,1987.15.K.Kedem,R.Livne,J.Pach,and M.Sharir.On the union of Jordan regions and collision-freetranslational motion amidst polygonal obstacles.Discrete Comput.Geom.,1:59–71,1986.mdan,J.T.Schwartz,and H.J.Wolfson.Object recognition by affine invariant match-ing.In Proc.IEEE put.Vision Pattern.Recogn.,pages335–344,1988.17.H.P.Lenhof and M.Smid.Sequential and parallel algorithms for the closest pairs problem.put.Geom.Appl.,5:273–288,1995.18. F.P.Preparata and putational Geometry:An Introduction.Springer-Verlag,New York,NY,1985.19.O.Schwarzkopf,U.Fuchs,G.Rote,and E.Welzl.Approximation of convexfigures by pairsof put.Geom.Theory Appl.,10:77–87,1998.20.M.Sharir.On-sets in arrangements of curves and surfaces.Discrete Comput.Geom.,6:593–613,1991.。