Rotational and line Symmetry
《轴对称图形》跨学科活动方案
《轴对称图形》跨学科活动方案English Response:Introduction:Cross-curricular activities that integrate different subject areas provide students with a comprehensive and engaging learning experience. One such activity thatfosters interdisciplinary connections is the exploration of symmetry, particularly in the context of axisymmetric shapes. This activity combines mathematical concepts, artistic expression, and critical thinking skills.Learning Objectives:Understand the concept of symmetry, specifically axisymmetric shapes and their transformations.Apply mathematical formulas and principles to determine the properties of axisymmetric shapes.Develop artistic skills by creating symmetrical designs inspired by mathematical concepts.Foster critical thinking and problem-solving abilities through the analysis and interpretation of symmetrical forms.Materials:Drawing paper.Markers or crayons.Ruler.Compass.Protractor.Copies of images or objects with axisymmetric shapes for observation and analysis.Procedure:Part 1: Mathematical Exploration.Introduce the concept of symmetry and axisymmetric shapes.Students use rulers, compasses, and protractors to draw and measure various axisymmetric shapes, such as circles, ellipses, spheres, and cones.They apply mathematical formulas to determine the radius, diameter, circumference, area, and volume of these shapes.Students investigate the properties of axisymmetric shapes, such as rotational symmetry and the relationship between the degree of symmetry and the number of axes of symmetry.Part 2: Artistic Expression.Inspired by the mathematical concepts explored, students create symmetrical designs on drawing paper.They use markers or crayons to draw patterns, shapes, and images that exhibit axisymmetric symmetry.Students can fold and cut paper to create three-dimensional symmetrical designs.Part 3: Critical Thinking and Analysis.Students observe and analyze images or objects with axisymmetric shapes.They identify the axes of symmetry, degree of symmetry, and any mathematical principles that apply to the shapes.Students present their findings to the class, discussing the mathematical and artistic aspects of the symmetrical forms.Assessment:Student drawings and designs.Class discussion participation.Written or oral presentations on the analysis of axisymmetric shapes.跨学科活动方案。
关于菱形的英语作文高中
关于菱形的英语作文高中标题，Exploring the Beauty and Mathematics of Diamonds。
Diamonds, those mesmerizing gems, have captivated human fascination for centuries. From their sparkling brillianceto their geometric perfection, diamonds are not onlyobjects of beauty but also subjects of mathematical intrigue. In this essay, we delve into the enchanting world of diamonds, exploring their aesthetics, structure, and mathematical properties.At first glance, a diamond appears as a simple shape –a rhombus with four equal sides. However, upon closer inspection, one discovers its symmetrical elegance and precision. Each angle measures 60 degrees, and opposite angles are congruent, making it a parallelogram. This geometric perfection is one of the defining features of diamonds, contributing to their allure in jewelry and design.Beyond their aesthetic appeal, diamonds possess remarkable mathematical properties. One of the most intriguing aspects is their symmetry. Diamonds exhibit various types of symmetry, including reflection symmetry and rotational symmetry. Reflection symmetry means that a shape can be divided into two identical halves by a line. In a diamond, this is evident when you draw a line from one corner to its opposite corner – the two resulting halves mirror each other perfectly. Rotational symmetry refers to the ability of a shape to be rotated by a certain angle and still look the same. A diamond has rotational symmetry of order two, meaning it looks the same when rotated by 180 degrees around its center. These symmetrical properties not only enhance the visual appeal of diamonds but also play a crucial role in their classification and evaluation in the gem industry.Moreover, diamonds are intricately connected to the field of geometry, particularly in the study of polyhedra.A diamond can be seen as a three-dimensional shape known as a rhombic dodecahedron. This polyhedron consists of 12 rhombus-shaped faces, with three meeting at each vertex.Its symmetrical structure and geometric properties have fascinated mathematicians and artists alike, inspiringworks of art and architectural designs.The mathematical fascination with diamonds extends beyond their shape and symmetry to their optical properties. Diamonds are renowned for their ability to refract and disperse light, resulting in their characteristicbrilliance and fire. This phenomenon is governed by the principles of physics and mathematics, particularly Snell's law of refraction and the theory of total internal reflection. By understanding these principles, scientists and gemologists can predict and enhance the optical performance of diamonds, ensuring their maximum beauty and value.In addition to their mathematical allure, diamonds have played a significant role in human culture and history. As symbols of wealth, power, and love, diamonds have adorned royalty, inspired poets, and fueled exploration and trade throughout the ages. From the legendary mines of Golcondato the modern laboratories producing synthetic diamonds,the story of diamonds is intertwined with human ingenuity, ambition, and creativity.In conclusion, diamonds are not merely dazzling gems but also fascinating objects of mathematical inquiry. Their symmetrical beauty, geometric perfection, and optical properties have captivated minds for centuries, inspiring exploration and admiration across cultures and disciplines. Whether adorning a piece of jewelry or sparking curiosityin a classroom, diamonds continue to shine as symbols of both natural wonder and mathematical elegance.。
symmetric 名词
Symmetric1. IntroductionSymmetric is a term used in mathematics and other fields to describe objects or concepts that possess a certain type of symmetry. In mathematics, symmetry refers to a property where an object remains unchanged under certain transformations, such as reflection, rotation, or translation. In this article, we will explore the concept of symmetry and its various applications in different disciplines.2. Symmetry in MathematicsSymmetry plays a fundamental role in mathematics and is widely studied in various branches, including geometry, algebra, and group theory. In geometry, symmetric objects are those that can be divided into two or more parts that are mirror images of each other. For example, a circle is symmetric with respect to any line passing through its center.Symmetry can also be observed in patterns and shapes. Regular polygons such as squares and equilateral triangles possess rotational symmetry because they can be rotated by certain angles without changing their appearance. Fractals, which are intricate mathematical patterns that repeat at different scales, often exhibit self-similarity and possess various types of symmetries.In algebra, symmetry is explored through the concept of functions. A function is said to be symmetric if it satisfies the condition f(x) =f(-x) for all values of x in its domain. This property implies that the graph of the function is symmetric with respect to the y-axis.Group theory provides a rigorous framework for studying symmetry by defining mathematical structures called groups. A group consists of a set of elements and an operation that combines two elements to produce another element in the set. Symmetry groups describe all possible symmetries of an object or system and play a vital role in understanding its properties.3. Symmetry in PhysicsSymmetry has profound implications in physics and is essential for understanding the laws of nature. The principle of symmetry lies at theheart of many fundamental theories, such as classical mechanics, quantum mechanics, and general relativity.In classical mechanics, the conservation of angular momentum is a consequence of rotational symmetry. The laws of motion remain the same regardless of the direction in which an object is oriented. This symmetry is evident in everyday life, where the behavior of objects under different orientations follows consistent patterns.In quantum mechanics, symmetry plays a crucial role in determining the behavior of particles and their interactions. The principles of quantum field theory rely heavily on the concept of gauge symmetry, which describes the invariance of physical laws under certain transformations. Symmetries such as time translation symmetry and particle-antiparticle symmetry are fundamental to our understanding of elementary particles and their interactions.General relativity, Einstein’s theory of gravity, incorporates the principle of general covariance, which states that physical laws should be independent of the choice of coordinates. This symmetry reflects the idea that space and time are not absolute but rather depend on the observer’s frame of reference.4. Symmetry in BiologySymmetry is prevalent in biology and plays a crucial role in understanding various biological processes and structures. Many organisms exhibit bilateral symmetry, where their bodies can be divided into two mirror-image halves along a plane. This type of symmetry is observed in animals ranging from insects to mammals.Bilateral symmetry provides advantages such as improved mobility and sensory perception. It allows for efficient movement by dividing body parts into corresponding pairs such as legs or wings. It alsofacilitates better coordination between sensory organs located on opposite sides of an organism.Symmetry is also observed at smaller scales within organisms. For example, DNA molecules possess helical symmetry due to their double-stranded structure. Proteins often exhibit internal symmetries that influence their folding patterns and functions.Studying symmetric patterns in nature can provide insights into evolutionary processes and ecological relationships between differentspecies. Understanding how symmetrical structures arise and function can help biologists unravel complex biological systems.5. Symmetry in Art and DesignSymmetry has been a fundamental principle in art and design for centuries. Artists and designers often use symmetrical patterns and compositions to create aesthetically pleasing and visually balanced works.Symmetry can be found in various art forms, including painting, sculpture, architecture, and textiles. The use of bilateral symmetry can create a sense of harmony and orderliness. Radial symmetry, where elements are arranged around a central point, is also frequently employed to achieve balance and visual interest.In modern design, the concept of symmetry has evolved to include asymmetrical compositions as well. Asymmetry introduces a dynamic element by intentionally breaking traditional symmetrical arrangements. This approach adds visual tension and can create a more engaging and thought-provoking experience for the viewer.6. ConclusionSymmetry is a fascinating concept that permeates many aspects of our world, from mathematics to physics, biology, and art. It represents an inherent orderliness and balance that is both aesthetically pleasing and intellectually stimulating.The study of symmetry has led to significant advancements in various fields, providing insights into the fundamental laws of nature, the structure of biological systems, and the principles of design. By understan ding symmetry’s underlying principles and exploring its applications, we gain a deeper appreciation for the intricate patterns that surround us.。
proe英语单词
Pro/E英语单词Navigation Area——浏览区Model tree——模型树Layer——图层File——文件View——视图Insert——插入Analysis——分析Info——信息Tools——工具Applications——应用程序Window——窗口Help——帮助Dashboard——仪表板Feature——特征Extrude——拉伸Revolve——旋转Sweep——扫描Blend——混合Hole——孔Shell——壳Rib——肋Draft——拔模Round——倒圆角Chamfer——倒角Constraint——几何限制条件Sketch plane——草绘平面Sketch orientation reference——草绘方向的参照Orientation——方向Depth——深度Standard Orientation ——标准方向Default Orientation——默认方向Placement——放置Section——截面Options——选项Capped ends——封闭端Properties——属性Material direction——材料方向Internal——内部Variable Section Sweep——可变剖面扫描Protrusion——伸出项Thin Protrusion——薄伸出项Cut——切口Thin Cut——薄板切口Surface——曲面Sketch Traj——草绘轨迹Select Traj——选取轨迹Merge Ends——合并终点Free Ends——自由终点Add Inner Faces——增加内部面No Inner Faces——无内部面Sweep Traj——扫描轨迹Setup New——新设置Filp——反向Okay——确定Top——顶部Bottom——底部Right——右Left——左Default——预设Quit——退出SKET VIEW——草绘视图Attributes——属性Done——完成Defined——已定义Use Previous——使用先前的Material Side——材料移除侧Inside section——截面内侧Cancel——取消Preview——预览Constant Section——恒定截面Cap ends——合并端Origin——原点Skect placement point——草绘放置点Tangency——相切Parallel——平行Regular Sec——规则截面Skectch Sec——草绘截面Straight——直的Smooth——光滑Toggle Section——切换曲面Feature Tools——特征工具General——一般Rotational——旋转的Project Sec——投影截面Blind——盲孔Boundary Blend——边界混合Edit——编辑Solidify——实体化Linear——线性Radial——径向Diameter——直径Coaxial——同轴Edge——边Offset——偏移Primary——主参照Shape——形状Radius——半径Angle——角度ISO——公制螺纹UNC——英制粗螺纹UNF——英制细螺纹Note——注释Thru Thread——全螺纹Thread——螺纹Exit Countersink——退出埋头孔References——参照Non-default thickness——非预设厚度Remove——移除Click here to add item——单击此处添加项目Draft surfaces——拔模曲面Draft hinges——拔模枢轴Pull direction——拔模方向Sets——设置Transitions——过渡Circular——圆形Rolling ball——滚球Through curve——通过曲线Pieces——段Details——细节Vertex——顶点Value——值Location——位置Attactment——附件Create end surfaces——创建结束曲面Spine——骨架Distance——距离Corner——角落点Corner Chamfer——拐角倒角Enter-input——输入Change——更改Apply——应用Type Keyword——键入关键字Find Now——立即查找Prehighlight——预选加亮Through All——穿透Erase——拭除Current——当前Copy——复制Paste Special——特殊粘贴Simple——简单Apply Move/Rotate transformations to copies——对副本应用移动/旋转变换Close——关闭Advanced reference configuration——高级参照配置Family Table——族表Look In——查找范围Switch Dimensions——切换尺寸Parameter——参数Merge Part——合并零件Filter——过滤Type——类型Instance Name——实例名Curve chain——曲线链Datum Plane——基准平面Display——显示Loop——环Accept——接受Save——保存Undo——撤销Original——原始Make Datum——产生基准Loop surfaces——回圈曲面Approximate——逼近Thru Points——经过点CRV OPTIONS——曲线选项Use Xsec——使用剖截面From Equation——从方程Quilt/Surf——面组/曲面Tweak——扭面Optional——可选的Cartesian——笛卡尔坐标Cylindrical——柱坐标Spherical——球坐标Csys——坐标系Chain——链Exact——确切Mirror——镜像Hide original geometry——隐藏原始几何Trim——修剪trimming object——修剪对象Intersect——相交Unlink——断开链接Along direction——沿方向Normal to surface——垂直于曲面Wrap——包络Destination——目的地Center——中心Ignore intersection surface——忽略相交曲面Trim at boundary——在边界Measurements ——量度To Vertex——至顶点Boundary——边界Measurement type——测量类型Open ends——开放终点Copy all surface as is——按原样复制所有曲面Swap—交换Exclude sueface and Fill holes——排除曲面并填充孔Copy as dependent——复制为从属性Join—连接Keep trimming sueface——保留修剪曲面Thin trim——薄修剪Controls——控制Extend——延伸Method——方式Tangent——切线Along Edge=Along Side edge——沿着侧边NormBnd=Normal To Boundary edge—垂直延伸边Automatic Fit——自动拟合Controlled Fit——控制拟合Create side surface——创建侧曲面Expand——延展Replace——替换Should the entities be aligned——点线是否对齐Environment——环境Isometric——等轴侧Constraints——约束Natural——自然Contents——内容Status——状态Pause——暂停Dimension——尺寸Scan Curve——扫描曲线Gtol——几何公差Symbol——符号Surace Finish——表面光洁度Datum Tag——基准标签Annotation——注释Annotation Element——注释元素Save Status——保存状态Reset Status——重置状态Unhide——取消隐藏Smart——智能Save a Copy——保存副本Thicken——加厚Edit Definition——编辑定义Relations——关系式Model Player——模型播放器Fix Model——修复模型Regenerate features—再生特征Show Dims——显示尺寸Display each feature——显示每个特征Model Tree Columns——模型树列Tree Filters——树过滤器Feat Subtype——特征子类型Designate Name——指定名称Width——宽度Ordel——顺序ppress——隐含Highlighted features——加亮特征Confirm——确认Parent/Chlid——父项/子项Child Handing——子项处理Global Reference Viewer——全局参照查看器Part—零件Actions——操作Object——对象Suspend——保留Setup Note——设置注释Pattern——阵列Conflict——冲突Explain——解释Resolve Sketch——解决草绘Failure Diagnostics——诊断失败Axis——轴Edit Definition——编辑参照The system could not construct the intersetion of part and feature.——系统不能构建零件和特征的交截Feature aborted——特征中止Failed to intersect part——交截零件失败Edit Parameters——编辑参数Increment——增量Table——表Placement——放置Toggle Construction——编辑构造Ungroup——分解组Diamona——菱形Square——正方形Triangle——三角形Spiral——螺旋Assembly——组件Drawing——绘图Manufacturing——制造Format——格式Report——报表Diagram——图表Layout——布局Markup——标记Sub-type——子类型Interchange——互换V erify——校验Process Plan——处理计划Common Name——公告名称Use default template——使用预设模板Component Placement——元件放置Connect——连接Mate——匹配Align——对齐Coincident——重合Allow Assumptions——允许假设Set Working Directory——设置工作目录View Manager——视图管理器Explode——分解Style——样式Simp Rep——简化表示No Cross Section——无剖面Plannar——平面Single——单一Top Level——顶级Show X-Hatching——显示剖切线Bill of Materials——材料清单Quantity——数量Contains——包含Specify Template——指定模板Empty——空Browse——浏览Size——大小Portrait——纵向Landscape——横向Inches——英寸Millimeters——毫米Origin——原点Scale——比例Trimetic——斜轴侧Pick From List——从列表拾取Lock View Movement——锁定视图移动Auxiliary——辅助Coordinate Dimension——坐标尺寸Sheet——页面Arrows——箭头Jog——角拐Snap Line——捕捉线Shared Date——共享数据Import——导入Height——高度Categories——类别View visibility——视图可见性Partial View——局部视图Clip——修剪Custom scale——定制比例Wireframe——线框Spacing——间距Modify Line Style——修改线体Phantom——剖视图Sheet Metal——钣金件Store Map——储存映射Line Font——线型Petrieve——检索Show By——显示方式Pick to ordinate——拾取基线Symmetry Line Axis——轴对称线Spline——样条Switch to orinate——切换到纵坐标Cleanup——整理Decimal Place——小数位数With Leader——带引线Text Style Gallery——文本样式库Character——字符Slant angle——斜角Justification——对齐Horizontal——水平Crosshatching——剖切线Margin——边距Nominal value——公称值Upper tolerance——上公差Lower tolerance——下公差Basic——基础Ipspection——检查Witnessline——尺寸界线Dual dimension——双重尺寸Changes to either the current feature's geometry or to some other feature havecaused the references for user-defined transition (No. 1) to be lost.The affected transitions must have their references redefined.Redefine the TRANSITIONS element for the failed feature, and select theTransitions indicated above for redefinition. Reselect lost references.Could not construct feature.Select an edge with CTRL key pressed to add to this set, or create a new set by selecting an edge or a surface.。
机械类英语词汇中英文对照
actuation活动，激励，动作
L
additive附加的，加成的，添加的，添加剂
lag缓缓而行，滞后,落后于,押往监狱，加上外套
adequately充分地
lagos拉各斯，尼日利亚首都
affordable提供得起的
lathe车床
aggregate合计的，集合的
layer层,层次
assembly组合,装配,部件，汇编
atmospheric大气的，空气的
machinability切削加工型，机械加工性能
augment增加，扩大
machineframe机座，机架
austenite奥氏体
machinery机器，机械
austenitize奥氏体化，使成奥氏体
machining机械加工
above all尤其是，最重要的是，首先是
J
abrasion磨损
jacket水套
abut邻接,依靠
jerky急拉的，急动的
accessory附件
journal bearing滑动轴承
accommodate使适应
K
acetylene乙炔，电石气
kerosene煤油
activity connection活动连接
predominant卓越的，支配的，主要的,突出的，有影响的
coolant冷却剂，冷冻机
preference优先选择
coolant冷却液
prehistoric史前的，很久以前的
cooperation合作,协作
preliminary预备的，初步的
coordinate坐标
pressurize增压，给……加压
cut edge剪切刃
Rotational symmetry
Make a shape with rotational symmetry of order 2
© T Madas
Make a shape with rotational symmetry of order 4
© T Madas
Make a shape with rotational symmetry of order 3
© T Madas
Rotational Symmetry Practice
© T Madas
Find the order of rotational symmetry for each shape:
46Βιβλιοθήκη 32© T Madas
Find the order of rotational symmetry for each shape:
120°
360 ÷ 3 = 120
© T Madas
For rotational symmetry of order 5:
We rotate the shape by
72°
72°
360 ÷ 5 = 72
© T Madas
Rotational Symmetry Practice
© T Madas
Shade one or more squares to create
6
3
none
© T Madas
Rotation and
Rotational Symmetry
© T Madas
We we start we the flag shape below We want to create a new shape with rotational symmetry We pick a point as the centre of rotation For rotational symmetry of order 2:
symmetry 英语解释
symmetry 英语解释《Symmetry: Unveiling the Beauty of Balance》Symmetry is a fascinating concept that touches upon the fundamental principles of balance, harmony, and beauty in the world around us. From the magnificent works of art to the intricate patterns in nature, symmetry plays a crucial role in creating visual appeal and evoking a sense of orderliness.Defined as a correspondence in size, shape, or arrangement on opposite sides of a dividing line or plane, symmetry is found in various forms throughout our existence. It exists in the realms of mathematics, biology, architecture, and even in the smallest particles that make up our universe. By examining the concept of symmetry, we can gain a deeper understanding of the fundamental organizing principles that shape our world.In nature, symmetry is a prevalent feature. From the petals of a flower to the branches of a tree, many living organisms exhibit a remarkable balance in their structures. The perfect radial symmetry of a sunflower, with its petals arranged uniformly around a central axis, is simply awe-inspiring. Similarly, the bilateral symmetry found in animals, where each side of the body mirrors the other, gives rise to the undeniable beauty of creatures such as butterflies and peacocks.In the realm of art and design, symmetry has always been highly valued. Throughout history, artists have utilized symmetry to create aesthetic masterpieces that captivate our senses. From the mesmerizing patterns on a Persian carpet to the architectural wonders of the Taj Mahal, symmetry is a key element in enhancing the sense of harmony and proportion. Michelangelo's famous painting on the ceiling of the Sistine Chapel is a testament to the power of symmetry in evoking a sense of grandeur.Moreover, symmetry plays a crucial role in mathematics and physics. Symmetrical shapes and patterns are deeply intertwined with geometric principles and mathematical formulas. This relationship can be seen in the symmetry of polygons, such as squares and circles, as well as in intricate fractal patterns. Even in the realm of physics, concepts like mirror symmetry and rotational symmetry are fundamental to understanding the laws that govern our universe.Symmetry not only captivates our senses but also holds profound philosophical implications. The quest for symmetry has driven scientists and researchers to uncover the fundamental truths behind the natural world. By seeking patterns and order, we strive to unlock the mysteries of the universe and gain a deeper appreciation for the beauty inherent in its balanced design.In conclusion, symmetry is a captivating concept that transcends disciplines and permeates every aspect of our lives. From the delicate patterns found in nature to the magnificent works of art, symmetry unveils the hidden beauty of balance and harmony. By embracing the power of symmetry, we can develop a greater appreciation for the world around us and gain insight into the fundamental principles that shape our existence.。
八年级上册轴对称知识点总结
八年级上册轴对称知识点总结As we dive into the concept of symmetry, it's important to understand that symmetry plays a crucial role in the world around us. Symmetry is not only pleasing to the eye, but it also helps us make sense of our environment. In mathematics, symmetry can be defined as a balance or proportion that is achieved through exact correspondence of form and arrangement on opposite sides of a dividing line or plane.当我们深入研究对称概念时，重要的是要理解对称在我们周围的世界中扮演着重要角色。
对称不仅令人赏心悦目，而且帮助我们理解我们的环境。
在数学中，对称可以被定义为通过在一个分割线或平面的对立面上形式和排列的精确对应而实现的平衡或比例。
In the context of eighth grade mathematics, students are introduced to the concept of reflection symmetry, also known as line symmetry or mirror symmetry. This type of symmetry occurs when one half of a figure is a mirror image of the other half across a line of symmetry. The line of symmetry is an imaginary line that divides the figure into two congruent parts.在八年级数学的语境中，学生们被引入了反射对称的概念，也被称为线对称或镜面对称。
固体物理第二章第二节对称性和布拉维格子的分类
P28-29表2.1给出了32个晶体学点群，为了 便于大家看懂，下面给出符号的说明
Cn C1, C2 , C3, C4 , C6
900 1200
900
7个晶系(crystal system)相应的点群 S1, C2h , D2h , D4h , D3d , D6h , Oh
即：Ai G,i 1, 2,3 ,G {Ai}
必须满足下列条件： 1). 封闭性(closure property) 按照给定的乘法规则，群G中任何两个元素 相乘，得到的还是该群的一个元素。
Ai Aj Ak ,i j or i j
2). 群中一定包含一个不变元素(单位元素) E
E G, EAi Ai E Ai
我们这里要讨论的主要是晶格(或点阵)的对 称性(symmetry of lattice).
在晶格这个物理系统中，一种对称性是指某些 要素互相等价，而用来描述晶格的要素，无非就 是：点、线、面。而保持这些要素等价的操作---对称操作有三种：平移、旋转、镜反射。假设 在某一个操作过后，点阵保持不变，也就是每个 格点的位置都得到重复，那么这个相应的平移、 旋转或镜反射操作就叫作一个点阵对称操作。其 中的点、线、面分别叫做对称中心、对称轴、对 称面----称为对称元素
比如：绕x轴的旋转，设转角为θ，则有：
x x
y
y
cos
z sin
z
y
sin
z
cos
a11 a12 a13 1 0
0
Line and rotational symmetry
Regular Pentagon
5
1
4
2
3
Order 6
Equilateral Triangle
An equilateral triangle has rotational symmetry of order ?
Regular Polygons
Equilateral Triangle
An equilateral triangle has rotational symmetry of order ?
3
2
1 Order 2
4
2
1
Order 3
Order 4
1
Rotational Symmetry
What is the order of rotational symmetry of the shape below?
Rotational Symmetry
What is the order of rotational symmetry of the shape below? 5
2
1 Order 2
2
1
Order 3
Rotational Symmetry
Order 1 3
The order of rotational symmetry that an object has is the number of times that it fits on to itself during a full rotation of 360 degrees.
Order 1
Order 2
基于拓扑优化的自行车一体轮设计
设计与分析♦Sheji yu Fenxi基于拓扑优化的自行车一体轮设计苏阳(巴斯夫冲国)有限公司，上海200137)摘要:采用基于SIMP变密度法的拓扑优化方法,对自行车轮组进行了轻量化设计。
综合考虑了自行车运行时的复杂受力情况，分别建立了轮组的静态和动态有限元模型，并对优化后的结构进行了校核。
研究结果表明，优化后一体轮的质量减轻了17.6%，无论静载还是碰撞工况都能满足强度要，了结构的行性。
关键词:拓扑优化;轻量化;一体轮0引言自行车是行和 的一必要装备。
自行车的主要重量集中在车架和车轮上，在运动过程中车轮动动，车轮重量，动量，动，过重的车轮重行的体力。
车轮了的轮轮轮一组，轮组”,各组用。
对车轮进行轻量化设计，采用强度工一体轮在结构的基上减轻重量并化装工车轮的在自行车行感受的时。
于轻量化的研究在结构工采用诸拓扑优化优化优化一优化方法合的设计方设计。
中，拓扑优化的方法，是在的到满足约束条件又使目标函数最优的布局方式[1]更多地应用于初概念设计阶段后再根据布局，结合具体要进行详细设计研究采用优化软TOSCA对车轮进行拓扑优化分析，在保证轮组静强度及抗冲击性能的前提下降低轮组的重量，实现一体轮的轻量化设计。
1拓扑优化方法及数学模型续体拓扑优化方法繁多，根据原理的不见拓扑优化方法均匀化法、变密度法、渐进结构优化方法(ESO)、水平集法[3#4]O中，变密度法将单元密度和联来，设计变量化相对密度，计算规模大。
概念单，算法于实现，优化效率高些是工应用中较为熟的拓扑优化方法用于O p tiStruct、TOSCA商业软件中。
目变密度法中最见的材料插值方法是带惩罚因子的固体向微结构模型(SIMP)和的合理近似模型(RAMP),两分别采用q形式的罚值和有理函数的方式来建立插值模型[6#8]公式(1)(2)所示。
SIMP模型：E=E>!⑴RAMP模型："1?(^⑵式中，E待的弹性模量；E。
原的弹性模量;。
模具专业英语词汇
模具专业英语词汇Aabrasion n. 磨损abrasion resistance n. 耐磨损性 abrasive n. 磨料 accelerator n. 促进剂 accuracy n. 准确性accurate die casting 精密压铸 air trap 积风 acrylic n. 丙烯酸 /压克力 ì active plate 活动板 additive n. 添加剂 adhere v. 黏附 adhesion n. 黏合 adhesive n. 胶粘剂air-cushion eject-rod 气垫顶杆 air cushion plate 气垫板 air entrapment n. 困气 anneal v. 退火 assemble v. 总成 Bback pressure 背压 bismuth mold 铋铸模 baffle plate 挡块 barrel n. 机筒 /料筒 / bending block 折刀 bottom block 下垫脚bottom plate 下托板(底板) bushing bolck 衬套barrel temperature 料筒温度 blush 发blank through dies 漏件式落料模 burnishblow molding n. 吹塑成型 blow molding machine n. 吹塑机 brittle adj. 脆性 bubble n. 气泡 burr 毛刺button dby-product n. 副产品 Ccalendering n. 压延 carbon steel n. 碳素钢 casting n. 铸造 catalyst n. 催化剂 cavity n. 型腔chemical resistance n. 耐化学腐蚀性 chip v. 削 /凿clamping force n. 锁模力 clamping tonnage n. 锁模吨位 coil spring 弹簧computer aided design (CAD) n. 电脑辅助设计computer aided manufacture (CAM) n. 电脑辅助制造cover plate盖板computer numerical control (CNC) n. 电脑数字控制ceramic n. 陶瓷calendaring molding 压延成形cavity 型控母模center-gated mold 中心浇口式模具clod hobbing 冷挤压制模center-gated mold 中心浇口式模具clod hobbing 冷挤压制模chill mold 冷硬用铸模cold chamber die casting 冷式压铸cold forging 冷锻cold slag 冷料渣cold slug 冷斑cold rolled steel n. 冷轧圆钢cold runner n. 冷流道colorant n. 着色剂compacting molding 粉末压出成形composite n. 复合材料composite dies 复合模具compound molding 复合成形compression molding 压缩成形compression strength n. 抗压强度cooling channel/circuit n. 冷却管道cooling rate n. 冷却速率core n. 型芯 /模芯corrosion n. 腐蚀corrosion resistance n. 耐腐蚀性counter punch 反凸模cure v. 固化custom adj. 定制cycle time n. 循环时间Ddegassing n. 排气density n. 密度deform v. 变形delamination 起皮diaphragm gate 隔膜浇口die n. 冲模die casting n. 压铸 /模铸deburring punch压毛边冲子die holder下夹板die pad下垫板die set下模座dimension n. 尺寸dimensional tolerance n. 尺寸公差dowel pin固定销dish gate 盘形浇口dip mold 浸渍成形distort v. 扭曲double stack mold 双层模具draft angle n. 拨模斜度drying 烘干duplicated cavity plate 复板模dwell time n. 驻留时间dye n. 染料Eedge gate 侧缘浇口ejector mark n. 顶出痕迹ejection pin n. 顶出杆/推顶锁 /脱模锁ejection plate n. 推顶板/脱模板ejection rod n. 脱模板拉杆elastomer n. 弹性体electrical discharge machining (EDM) n. 电火花加工electroformed mold 电铸成形模encapsulation molding 注入成形epoxy n. 环氧树脂eq-height sleeves=spool等高套筒expander die 扩径模extruder n. 挤出机extrusion n. 挤出，挤塑extrusion die 挤出模extrusion molding 挤出成形Ffabricate v. 制造family mold 反套制品模具fan gate 扇形浇口fantail die 扇尾形模具fiber n. 纤维feature die公母模female die母模(凹模)fiberglass n. 玻璃纤维fiber reinforcement n. 纤维增强film n. 薄膜film gate 薄膜浇口fire retardant n. 阻燃剂fishtail die 鱼尾形模具flash 飞边/溢料flash gate 闸门浇口flash mold 溢料式模具flaw/scratch 刮伤flexural modulus 弯曲模数flexural strength 抗弯强度flow line 流痕flow mark n. 流痕flow rate n. 流动速率fluorescent adj. 荧光fluorescent brightener n. 荧光增白剂foam n. 泡沫fuse v. 熔合Ggang dies 复合模gas assisted injection molding n. 气辅注射成型gate n. 浇口gate design 浇口设计gate mark n. 浇口痕gear n. 齿轮grinder n. 研磨机 /磨床gas mark 烧焦glazing 光滑gloss 光泽foam forming 发泡成形forming die 成型模forging roll 轧锻gravity casting 重力铸造groove punch压线冲子guide pin导正销guide plate定位板guide pad导料块gypsum mold 石膏铸模H heat conduction n. 热传导heater band n. 加热圈heat treatment n. 热处理hesitation 迟滞high density polyethylene (HDPE) n. 高密度聚乙烯high impact plastic n. 高抗冲塑料high impact polystyrene n. 高抗冲聚苯乙烯hobbing n. 滚铣/挤压制模 /切压制模holding n. 保压hollow(blow) molding 中空(吹出)成形hot runner n. 热流道hot chamber die casting 热室压铸hot forging 热锻hot-runner mold 热流道模具Iimpact strength n. 冲击强度ingot mold 钢锭模injection n. 注射/注塑injection mold 注塑模injection molding 射出成形injection molding machine n. 注塑机injection pressure 注塑压力injection speed 注塑速度inner stripper内脱料板inner guiding post内导柱inner hexagon screw内六角螺钉insert入块(嵌入件)insulated runner n. 绝热流道/保温流道insulation layer n. 绝热层internally heated runner n. 内加热流道internal void n. 内部空腔investment casting 精密铸造Jjoint n. 接头/连接/Llancing die 切口模laminating method 被覆淋膜成形lancing die 切口模landed plunger mold 有肩柱塞式模具landed positive mold 有肩全压式模具laser n. 激光laser cutter n. 激光切割机lathe n. 车床lifter guide pin浮升导料销lifter pin顶料销located block定位块located pin定位销loading shoe mold 料套式模具loose detail mold 活零件模具long nozzle 延长喷嘴方式loose mold 活动式模具louvering die 百叶窗冲切模lost wax casting 脱蜡铸造low density polyethylene n. 低密度聚乙烯low pressure casting 低压铸造lower plate下模板lower sliding plate下滑块板lower stripper下脱料板Mmale die公模(凸模)main runner 主流道manifold die 分歧管模具melt v. 熔融melt index n. 熔融指数mill (metal) v. 铣machining n. 机加工machining path n. 加工路径manufacture v. 制造matched die method 对模成形法matched mould thermal forming 对模热成形模meld line n. 融合线metal machining n. 金属加工melt temperature 熔化温度milling machine n. 铣床model n. 模型modular mold 组合式模具monomer n. 单体mold n. 模具mold cavity n. 模具型腔mold life n. 模具寿命mold temperature 模具温度molding conditions 成型条件modular mold 组合式模具multi color injection molding n. 多色注塑multi-cavity mold 多模穴模具multi-gate mold 复式浇口模具multi shot injection molding n. 多次注料注塑Nnozzle n. 喷嘴Nylon n. 尼龙Ooffset bending die 双折冷弯模具outer bush外导套outer guiding post外导柱outer stripper外脱料板oxidation resistance n. 耐氧化性oxidize v. 氧化Ppackaging n. 包装packing n. 补料palletizing die 叠层模parison n. 型坯part n. 零件parting line n. 分模线parting plane n. 分模面pierce die 冲孔模pin销pinhole gate n. 针孔形浇口pinpoint gate 点浇口plain die 简易模plaster mold 石膏模plastic n. 塑料plasticity n. 塑性plasticizer n. 增塑剂plotter n. 绘图机polish v. 抛光porous adj. 多孔precise adj. 精密polycarbonate (PC) n. 聚碳酸酯polyethylene n. 聚乙烯polyester n. 聚酯polymer n. 聚合物（体） /高聚物polymerization n. 聚合polystyrene n. 聚苯乙烯polyurethane n. 聚氨脂polyvinyl chloride (PVC) n. 聚氯乙烯porous mold 通气性模具positive mold 全压式/挤压式模具pressure die 压紧模profile die 轮廓模progressive die 顺序模portable mold 手提式模具positive mold 全压式模具powder forming 粉末成形powder metal forging 粉末锻造precision n. 精度precision injection machine n. 精密注塑机precision forging 精密锻造precision molding n. 精密模塑preform n. 预成型preform molding n. 预成型模塑press forging 冲锻pressure die 压紧模profile die 轮廓模progressive die 连续模/顺序模prototype mold 雏形试验模具process v. 处理chǔlǐ；加工production line n. 生产线prototype mold 雏形试验模具prototype n. 原型punch冲头punch set上模座punching die 落料模punch holder上夹板punch pad上垫板Qquench v. 淬火Rraising(embossing) 压花起伏成形re-entrant mold 倒角式模具reel-stretch punch卷圆压平冲子regrind n. 回用料regrind usage 次料使用resin n. 树脂rib n. 肋条rib stiffener n. 加强肋ribbon punch压筋冲子rigidity n. 刚性ring gate 环形浇口ripple n. 波纹/皱纹riveting die 铆合模rocking die forging 摇动锻造roller滚轴rotary forging 回转锻造rotational molding 离心成形rough adj. 粗糙round punch圆冲子rubber n. 橡胶 /橡皮írubber molding 橡胶成形runner n. 流道runner design 流道设计runner plat 浇道模块runner system浇道系统runnerless adj. 无流道runnerless injection mold n. 无流道注塑模具Ssand mold casting 砂模铸造scale n. 比例schematic n. 图解screw speed 螺杆转速seal v. 密封secondary runner 次流道sectional die 拼合模/对合模具segment mold 组合模semi-positive mold 半全压式模具shaper 定型模套shearing die 剪边模shell casting 壳模铸造short shot n. 短射shot n. 注料shot size n. 注塑量shrinkage 收缩(率)side core n. 侧性芯side gate 侧浇口silicon n. 聚硅氧烷silicon rubber n. 硅橡胶simulate v. 模拟single cavity mold 单腔模具single gate mold n. 单浇口模具sink mark n. 凹痕 /缩痕sinter forging 烧结锻造six sides forging 六面锻造slag well 冷料井slide n. 滑动阀sliding block滑块sliding dowel block滑块固定块special shape punch异形冲子spring box弹簧箱spring-box eject-rod弹簧箱顶杆spring-box eject-plate弹簧箱顶板stop screw止付螺丝stripper pad脱料背板stripping plate内外打(脱料板)supporting block for location定位支承块slit gate 缝隙浇口slush molding 凝塑成形softener n. 软化剂solid forging die 整体/拼合锻模splay 银纹split forging die 拼合锻模split mold 分割式模具/双并式模具sprue 注道/ 浇口/溶渣sprue/cold material trap 浇道/冷料井sprue gate 射料浇口，直浇口sprueless mold 无注道残料模具sprue lock pin 料头钩销(拉料杆)sprue puller 拉杆sprueless mold 无射料管方式/无注道残料模具squeeze casting 高压铸造squeezing die 挤压模steam channel n. 汽道stabilizer n. 稳定剂stamped punch字模冲子stamping n. 冲压stainless steel n. 不锈钢stiffening rib punch = stinger 加强筋冲子strain n. 应变streak n. 条纹 /条痕stretch form die 拉伸成形模stress n. 应力stripper n. 脱模器/顶出器stripper ring n. 脱模圈submarine gate 潜伏浇口surface defect n. 表面缺陷surface treatment n. 表面处理surface check 表面裂痕swing die 振动模具symmetrical adj. 对称symmetry n. 对称swaging 挤锻sweeping mold 平刮铸模Ttab gate 搭接浇口Teflon n. 特氟隆temper v. 回火tensile strength n. 抗张强度tensile elongation 延伸率thermoforming n. 热成型thermoplastic n. 热塑性塑料 adj. 热塑性thermoset n. 热固性塑料 adj. 热固性three-dimensional adj. 三维空间three plates mold 三片式模具tie bar n. 连接杆tolerance n. 公差tool mark n. 刀痕tool steel n. 工具钢top plate上托板（顶板）top block上垫脚transfer molding 转送成形trimming die 切边模trimming punch切边冲子tunnel gate 隧道式浇口Uultrasonic welder n. 超声波焊接机undercut n. 凹孔unit cost n. 单价unit mold 单元式模具universal mold 通用模具unscrewing mold 退扣式模具up stripper上脱料板upper plate上模板upper holder block上压块upper mid plate上中间板Vvacuum n. 真空valve gate阀门浇口vented injection molding n. 排气式注塑成型viscosity n. 黏度/黏性void 缩孔vulcanize v. 硫化Wwarm forging 温锻warpage n. 翘曲变形wear n. 磨损wear resistance n. 耐磨损性weld line n. 熔接线wire spring圆线弹簧working drawing n. 工程图yoke type die 轭型模机械设备:3D coordinate measurement 三次元量床boring machine 搪孔机contouring machine 轮廓锯床copy grinding machine 仿形磨床cylindrical grinding machine 外圆磨床die spotting machine 合模机driller 钻床EDM=Electron Discharge Machining 放电加工electrical sparkle 电火花engraving machine 雕刻机engraving E.D.M. 雕模放置加工机form grinding machine 成形磨床graphite machine 石墨加工机grinder 磨床horizontal boring machine 卧式搪孔机horizontal machine center 卧式加工制造中心internal cylindrical machine 内圆磨床lathe车床linear cutting 线切割miller 铣床milling machine 铣床planer |刨床punching machine 冲床robot机械手welder 电焊机分类（汉英）一、入水：gate进入位：gate location水口形式：gate type大水口：edge gate细水口： pin-point gate水口大小：gate size转水口：switching runner/gate 唧嘴口径：sprue diameter二、流道: runner热流道：hot runner,hot manifold 热嘴冷流道: hot sprue/cold runner 唧嘴直流: direct sprue gate 圆形流道：round(full/half runner 流道电脑分析：mold flow analysis 流道平衡:runner balance热嘴：hot sprue热流道板：hot manifold发热管：cartridge heater探针: thermocouples插头：connector plug插座： connector socket密封/封料： seal三、运水：water line喉塞：line plug喉管：tube塑胶管：plastic tube快速接头：jiffy quick connector plug/socker 四、模具零件：mold components三板模：3-plate mold二板模：2-plate mold边钉/导边：leader pin/guide pin边司/导套：bushing/guide bushing中托司：shoulder guide bushing中托边L：guide pin顶针板：ejector retainner plate托板：support plate螺丝： screw管钉：dowel pin开模槽：ply bar scot 撑头: support pillar唧嘴： sprue bushing挡板：stop plate定位圈：locating ring锁扣：latch扣鸡：parting lock set推杆：push bar栓打螺丝：S.H.S.B顶板：eracuretun活动臂：lever arm分流锥：spure sperader 水口司:bush垃圾钉：stop pin隔片：buffle弹弓柱：spring rod弹弓:die spring中托司：ejector guide bush中托边：ejector guide pin镶针：pin销子：dowel pin波子弹弓：ball catch喉塞: pipe plug锁模块：lock plate斜顶：angle from pin斜顶杆：angle ejector rod尼龙拉勾：parting locks活动臂：lever arm复位键、提前回杆：early return bar 气阀：valves斜导边：angle pin五术语：terms承压平面平衡：parting surface support balance模排气：parting line venting回针碰料位：return pin and cavity interference模总高超出啤机规格：mold base shut hight顶针碰运水：water line interferes withejector pin料位出上/下模：part from cavith (core) side模胚原身出料位：cavity direct cut on A-plate,core direct cut o n B-plate.内模管位：core/cavity inter-lock顶针：ejector pin司筒：ejector sleeve司筒针：ejector pin推板：stripper plate缩呵：movable core,return core core puller 扣机（尼龙拉勾）：nylon latch lock斜顶：lifter模胚（架）： mold base上内模：cavity insert下内模：core insert 行位（滑块）： slide镶件：insert压座/斜鸡：wedge耐磨板/油板：wedge wear plate压条：plate不准用镶件： Do not use (core/cavity) insert 用铍铜做镶件： use beryllium copper insert初步（正式）模图设计：preliinary (final) mold design反呵：reverse core弹弓压缩量：spring compressed length稳定性好：good stability,stable强度不够：insufficient rigidity均匀冷却：even cooling扣模：sticking热膨胀：thero expansion公差：tolorance铜公(电极）：copper electrod根据国家标准，以下为部分塑料模具成形术语的标准翻译。
做辅助线的原则（Theprincipleofdoingauxiliaryline）
做辅助线的原则（The principle of doing auxiliary line）Hello.A lot of people to do the auxiliary line for many years, did not want to do what is the purpose of the auxiliary line, in fact, the auxiliary line, is to establish the relation between the known.Method of auxiliary line of a variety of specific problems to specific analysis, but also has his own routines. This is I help you find from other places, very comprehensive summary.(1) in the triangle:The isosceles Delta: often on the bottom line or even high angle bisector or rectangular structure (nature Delta, two congruent or facilitate the use of a three line one isosceles. Figure 1)Have the right point: even the midline (delta hypotenuse construct two isosceles Delta, or to use special properties on the midline of the hypotenuse angle. Figure 2)The oblique delta midpoint or midline midline structure (two: even the bottom with high equalarea. Figure 3); or about two vertices are the vertical midline (two congruent triangle structure. Figure 4); or even the median line, or a point parallel line on the other side of the structure (two similarity ratio is similar to a 1:2, or for use in a delta line theorem. In Figure 5, 6); double or extended the median line or line (constructed two congruent or delta completion for a parallelogram. In Figure 7, 8). Or extend the midline of the1/3 (structure two congruent or complete to a parallelogram. Figure 9).The angle bisector: on the top of a node for both sides of the vertical angle (angle delta two congruent structure. Figure 10) parallel lines or one or both structure (one or two or a diamond isosceles. Figure 11).It is the angle bisector angle: this side from a section of the other side is equal to the vertex and the related connection structure (two congruent. In Figure 12, 13)We have met the vertical bisector: often extended vertical structure (isosceles delta. Figure 14).(two) the ladder:The extension of the two waist at a point (two a similar structure. Figure 15),By the end of a small bottom line parallel to the waist (two waist and construct a focus on two at the end of the poor and a parallelogram. Figure 16).The ends of a small bottom for the vertical sole (two square and rectangular structure. Figure 17).The diagonal line: by the end of the small bottom line parallel to another diagonal structure (a concentrated two diagonal and two bottom and the Delta and a parallelogram. Figure 18).The bottom end and the other with a small waist and waist point extension and intersection (both structural and a trapezoid and a higher product. Figure 19).The midpoint of a waist line parallel another waist (parallelogram structure such as delta and co-existence of trapezoid product. Figure 20).The point is too small at the end of the two parallel lines were constructed from the waist (two waist and two at the end of the poor and the delta two parallelogram. Figure 21).(three round):There is even a string: String vertex radius, even perpendicular to chord diameter or chord distance (structural angle Delta, facilitate the use of vertical theorem, Pythagorean theorem, acute triangle function problem solving); or tangent chord end point and the center angle, circumferential angle (easy to use angle theorem. Figure 22).The diameter and vertical diameter of chord or half chord, chord connection endpoint and the diameter of the structure (three similar to a rectangular, nature and projective theorem by using rectangular delta. Figure 23).It is inscribed quadrilateral diagonal (circular structure: even more equal angles. Figure 24); or the extended quadrilateral (a side structure and inner diagonal equal angle. Figure 25).The tangent circle: even point radius or diameter (vertical tectonic relationship); or made a point of string and related central angle, circumferential angle (easy to use angle theorem. Figure 26).The outer circle has two intersecting tangent: even tangent radius, and tangent intersection with the center line (right triangle structure two congruent);Or as the intersection point and the secant (to facilitate the use of tangent secant theorem); two point (or link structure of a delta, Delta, three isosceles right angle to be congruent line perpendicular bisector tangent intersection and the center of the string, easy to use, right angle, isosceles Delta Delta Delta and congruent projection theorem. Figure 27).The intersect chord or intersect at the outer circle of the secant tangent intersection \: connecting the endpoints of different strings or different in the circle on the secant (structural similarity Delta, facilitate the use of the proportion of segments and delta angle theorem. In Figure 28, 29, 30).The two circle intersection: connecting line, public string, even to the ends of the two center point of public string connection (structure twoA kite, a isosceles complete, easy to use one line perpendicular bisector of common chord theorem. Figure 31).The two circle tangent: the connecting line and the inner andGrandpa tangent, even cut and even a centralized structure (radius two strings and Grandpa tangent lengthThe right angle, a right angle trapezoidal delta from two round radius, radius and Grandpa tangent length. Figure 32).To the two circle is inscribed: connecting line and Grandpa tangent (for vertical relationship by connecting line with common tangent. Figure 33).The two round: from outer connecting line and a common tangent or internal common tangent and small circle, parallel line structure (a common tangent angle on heart length, the common tangent length, radius difference or two. In Figure 34, 35).Figure 1 is known as long line, line to line.Rotational structure congruent, such as line angle substitution.Multi line even midpoint can be obtained in the bit line.If you know the angular bisector, both sides are vertical.Can also go along the fold, congruent figures show up.If the vertical bisector of an isosceles triangle, visible.Bisector with parallel line segment angular position change etc..The known line connecting both ends of line perpendicular bisector, etc..2 people say that geometry is difficult, difficult in the auxiliary line.The auxiliary line, how to add? To grasp the concept and theorem.But hard work and study, find out the rules by experience.A bisector in the title, to make both sides of the vertical.Line perpendicular bisector, can connect the wires to the ends.Two point triangle, links into the bit line.A midline triangle, extend the same long line.In proportion, are similar, often for parallel lines.If all the outer circle line, tangent circle to line.If the two round and cut through the tangent point as tangent.The two circles intersect at two points, the general public for its string.Is the diameter of a semicircle, want to do the right line.As a circle, isometric, and less difficult problem that.The auxiliary line is dotted, drawing attention does not change.A bisector graph, which makes vertical to both sides.Look at the map can also be folded, symmetrical after the relationship between the current.The angle bisector parallel lines, isosceles triangle to tim.The angle bisector with vertical, three in one try.Line perpendicular bisector, often to the ends of the line.To permit line and half times, shorten the test can be extended.Two point triangle, connecting into the bit line.A midline midline midline extension triangle.Parallelogram, center of symmetry points.As the translation of a trapezoidal wire, waist try.Parallel moving diagonal fill common triangle.Card similar than line segments, Tim line parallel to become a habit.The plot ratio formula is the key segment for change.Direct evidence of difficulties, equal replacement less trouble.The hypotenuse above line, a large proportion of items.The radius and length calculation, string distance to the intermediate station.If all the circle line, tangent circle radius.Calculate the tangent length of the Pythagorean theorem is the most convenient.In order to prove the tangent, radius of vertical resolution carefully.Is the diameter of a semicircle, at right angles to the string size even.Arc point center, vertical theorem to remember all.Circular corner two strings, the diameter of chord endpoint even.The edge angle tangent chord, with diagonal arc find end.If you want to make a circle, make each side ".Make a circle, inner angularbisector dreamIf the intersection circle, don't forget to do public string.Inside and outside the two tangent circle through common tangent.If you add in the above point connecting line, sure.To add a equiangular circle, proved less difficult problem.The auxiliary line is dotted, drawing attention does not change.If the graph is scattered, symmetrical rotation to experiment. Basic mapping is the key, usually to grasp skilled. Problem solving more seriously, often summarize method.Do not blindly add lines, flexible methods should be varied.The comprehensive analysis methods, the more difficult will be less.Diligent modestly and diligence, scores rise straight。
有限元常用命令英文汉译
有限元常用英文汉译File（文件）New----新建Save----保存Close----关闭Open----打开Save as----另存为（当前模型也更名为）Save to----另存为（当前模型不更名为）Browse----浏览Import/Export----输入/输出（支持MSC-Nastran, .dat .nas文件,AutoCAD DXF, .dxf文件, ACIS, .sat文件, Stero-listography, .stl文件, Bitmap，可直接编辑文本文件.txt）Save sub model----存储子模型Make beam section----建立梁截面特性文件Make library----建立材料特性库或截面特性库Print setup----打印设置Preference----预置：Scratch files----临时文件路径Clean----清理临时文件Property libraries----特性库Default units----缺省单位制Auto save (minutes)----自动存储时间间隔Backup files on open----后备文件打开名Inspector font----实体检查字体Show full group----显示整个组Toggle select: Additive/Exclusive----锁定选择：复选/单选Exit----退出Edit （编辑）:Node----节点Element----单元Geometry----几何Select----选择：select all----全选，by region----按区域选，by property----按特性选by group----按组选clear all select----清除全部选择Cut----剪切Copy----复制Paste----粘贴Delete----删除Find----查找Undo----取消Redo----重做Online editor----在线编辑View:（显示）Redraw----重画Dynamic----动态显示Fresh----刷新Clear----清理Zoom----缩放：In----将指定区域图象放大到整个屏幕；Out----将整个图象缩至指定区域内；Last----上一次Draw----画Pan----平移Scale----按比例显示Angles----视角MultiView----多个模型同时显示Snap grid----捕捉栅格Show/Hide----显示/隐藏Display----显示选择node----节点, beam----梁, plate-----板, brick----块, links----连接, node attributes----节点属性, element attributes----单元属性,vertices----角节点, geometry----几何Show by property----按特性显示Beam free ends----梁自由端Plate free edges----板自由边Brick free edges----块自由面Element display----单元显示Attribute display----属性显示Options----选项Axes & screen Tab----关于轴和屏幕,Dynamic Rotation Tab----动态显示,Drawing Tab----绘图,Numbers Tab----数字显示,Free Edge Tab----自由边显示,Selecting Tab----关于选择Toolbars----工具条Entity toggles----实体锁定Summary:（摘要）Information----模型信息Total----总数,Units----单位,Load cases----载荷工况,Freedom cases----自由度工况,Properties----材料特性,Table----表,Comments----评述Property----特性信息Beam 梁，Plate 板，Brick 块Model----模型Bill of material 材料清单（具有这种材料的单元实体等数量等情况）Mass distribution质量分布(质心)Local and Global Mass Moments of Inertia----局部坐标系下\总体坐标系下的惯性矩等截面特性。
做对称的描述方式
做对称的描述方式我们首先来看看，wiki上是怎么给具体的几何对称下定义的：A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation.你看，几何对称性最粗浅的认识，就还真是两个对象的对应性，如最典型的轴对称，就是所谓的翻折重合。
但实际上，这个对象的数量可以更多，变换方式也可以多种多样，物理上合理即可。
更重要的是，其更本质的特点是对称不变性，即以集合方式描述的像素点级别的不变性。
而对应性仅仅是对称不变性的低阶特点而已，比如中心对称在中小学阶段比轴对称稍微难理解，就在于它更适合用旋转180度不变这样的对称性，而不是两个部分转180度互相重合的对应性来描述更符合直觉。
进一步来理解，一个几何对称图形一定是可以拆分成若干个部分的并集的，这个集合的元素就是群内元素，而且某个几何操作恰好能够生成它们，构成生成群。
所以才有了，几何对称图案，其实是基础图案加上具有某性质的群操作构成的。
这一切都符合上一讲中介绍的生成群的概念。
那么在几何对称中，我们自然要研究的就是，对于一个基础图案，有哪些满足各种性质的双射几何操作，以及我们肉眼已见的图案，分别又可以看作哪些操作生成的呢？空间几何对称类型介绍1.反射对称(reflectionalsymmetry)：也叫线/轴对称（linesymmetry）或镜像/面对称（mirrorsymmetry），一般以2维还是3维对象来区分，自然也可以抽象到n维空间去，只是那就进入抽象几何的领域了。
symmetrical
symmetricalSymmetricalIntroductionSymmetry is a fundamental concept in various fields of science, art, mathematics, and nature itself. It refers to a balanced arrangement of parts on either side of a central axis, resulting in a mirror image. The concept of symmetry has intrigued humans for centuries and has been used to create aesthetically pleasing designs, to study patterns in nature, and to understand fundamental principles in various disciplines. In this document, we will delve into the concept of symmetry, explore its different forms, discuss its significance, and look at some real-world examples.Understanding SymmetrySymmetry can be defined as a balanced and harmonious arrangement of parts. It can be observed in different aspects of our daily lives, ranging from the human body to plants, animals, and even inanimate objects. The concept ofsymmetry can be categorized into different types based on the nature of the objects being observed.Types of Symmetry1. Reflectional Symmetry: Also known as mirror symmetry, reflectional symmetry occurs when an object can be divided into two equal halves along a line or a plane. The two halves are mirror images of each other. Examples of objects exhibiting reflectional symmetry include butterflies, human faces, and buildings that have symmetrical facades.2. Rotational Symmetry: Rotational symmetry occurs when an object can be rotated around a central point and still maintains its original appearance. The object may have multiple axes of rotational symmetry. Examples of objects with rotational symmetry include wheels, flowers like sunflowers, and snowflakes.3. Translational Symmetry: Translational symmetry occurs when an object can be shifted along a specific direction and still maintains its original appearance. This type of symmetry is commonly observed in patterns such as wallpaper designs, textiles, and some architectural elements.4. Rotational-Reflectional Symmetry: This is a combination of both rotational and reflectional symmetries. An object with rotational-reflectional symmetry can be both rotated and reflected to create a perfect match. Examples include snowflakes and many intricate geometric designs.The Significance of SymmetrySymmetry holds great importance in various disciplines, from mathematics and physics to biology and art. Here are some ways symmetry influences these fields:1. Mathematics: Symmetry is an essential concept in the field of mathematics, exhibiting deep connections with geometry, group theory, and topology. Symmetry is used to study patterns, shapes, and fundamental principles in these areas.2. Physics: Symmetry plays a crucial role in physics, particularly in quantum mechanics and particle physics. The laws of physics are often described by symmetrical equations and principles. Symmetry has aided in predicting the existence of new particles and understanding the behavior of fundamental forces.3. Biology: Symmetry is evident in the natural world, from the bilateral symmetry of animals like insects and humans to the radial symmetry of plants like flowers. It is believed that symmetrical features in organisms are associated with better health and genetic fitness.4. Art and Design: Symmetry has been utilized by artists and designers throughout history to create aesthetically pleasing compositions. It is seen in various art forms, architectures, and even in the human perception of beauty. Symmetry provides a sense of balance and harmony in visual compositions.Real-World Examples of Symmetry1. Taj Mahal: The iconic Taj Mahal in India is known for its symmetrical design. The mausoleum is perfectly mirrored along a central axis, creating a stunning reflection in the surrounding water.2. Butterfly Wings: The intricate patterns on butterfly wings often exhibit reflectional symmetry. The wings are divided into two equal halves, each mirroring the other.3. Snowflakes: Snowflakes are renowned for their symmetrical and intricate structures. Each snowflake exhibits both rotational and reflectional symmetries, making them unique and beautiful.4. The Sunflower: The spiral arrangement of seeds in a sunflower exhibits rotational symmetry. The seeds form concentric circles, giving the flower a harmonious and balanced appearance.ConclusionWhether found in nature, mathematics, or art, symmetry is an intriguing concept that captivates our attention. The different types of symmetry, including reflectional, rotational, and translational, are present in various aspects of our lives. Symmetry plays a crucial role in understanding the world around us, from predicting the behavior of subatomic particles to appreciating the beauty of a natural flower. As we continue to explore the wonders of symmetry, we gain deeper insights into the principles that govern our universe and the underlying harmony that exists in our surroundings.。
Deform软件中对象设置翻译与设置
Deform软件中对象设置翻译与设置源自：deform论坛GeneralObject Type——对象类型，包括以下五种：Rigid——刚性体，通常指不变形材料，比如模具就会被定义为刚体；Plastic——塑性体，主要指刚塑性材料，其应力随应变率呈非线性增加，直到应变率极限，通常用来定义坯料（Elastic——弹性体，弹性材料的性能可由弹性和泊松比来表现，常用于求解模具的应力或挠曲线；Porous——多孔体，除了仿真过程中需要求解和更新材料密度的情况以外，可以将多孔体视为塑性体一处理；Elasto-plastic——弹塑性体，弹塑性体在达到材料屈服点以前，可以视为弹性体，当对象的某部分达到屈服点后而其余部分仍被视为弹性体。
Temperature——设置对象初始温度；Material——给对象定义材料类型；Primary Die——是否将该对象定义为主模具，主模具就是主导运动的模具。
Geometry这个选项卡主要是用于输入几何模型（Import Geometry）并对其进行检查核对等操作，对于一般的三维模型，保存为STL格式，基本就能直接输入而不用进行太多的调整。
如果是有对称面的刚性体（模具），需要在Symm 选项卡下输入对称面。
MeshTetrahedral mesh——四面体网格，DEFORM只能划分四面体风格，但是可以通过Import mesh进去输入别的网Brick mesh——四边形网格，二维模拟所用的网格类型；Number of elements——网格数量，只有选择相对网格划分时滑动条才有效；Preview——预览（网格划分情况）；Generate mesh——生成网格；Check Mesh——检查网格，点击可看到最小单元边界长度，对确定步长有很大帮助。
绝对网格和相对网格控制选中相对网格控制时，只需设定Numberof Elements里面的网格数就行，这和前面的网格滑动条是一致的。
圆锥曲线论 英文
圆锥曲线论英文Conic sections are a fundamental topic in mathematics that deals with the properties and equations of curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola. In this document, we will explore the characteristics and equations of these conic sections.1. Circle:A circle is a conic section formed when a plane intersects a cone at a right angle to its axis. It is defined as the set of all points in a plane that are equidistant from a fixed center point. The equation of a circle with center (h, k) and radius r is given by (x h)^2 + (y k)^2 = r^2.2. Ellipse:An ellipse is formed when a plane intersects a cone at an angle that is less than a right angle. It is defined as the set of all points in a plane for which the sum of the distances from two fixed points (called foci) is constant. The equation of an ellipse with center (h, k), major axis length 2a, and minor axis length 2b is given by ((x h)^2 / a^2) + ((y k)^2 / b^2) = 1.3. Parabola:A parabola is formed when a plane intersects a cone parallel to one of its generating lines. It is defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). The equation of a parabola with vertex (h, k) and focal length p is given by (x h)^2 = 4p(y k).4. Hyperbola:A hyperbola is formed when a plane intersects a cone at an angle greater than a right angle. It is defined as the set of all points in a plane for which the absolute value of the difference of the distances from two fixed points (called foci) is constant. The equation ofa hyperbola with center (h, k), transverse axis length 2a, and conjugate axis length 2b is given by ((x h)^2 / a^2) ((y k)^2 / b^2) = 1.These equations provide a mathematical representation of the conic sections and allow us to analyze their properties. By manipulating these equations, we can determine important characteristics such as the shape, size, orientation, and position of the conic sections.In addition to their geometric properties, conic sections have various applications in different fields. For example, circles are commonly used in geometry, physics, and engineering to represent objects with rotational symmetry. Ellipses are used in astronomy to describe the orbits of planets and satellites. Parabolas are used in physics to model the trajectory of projectiles and in engineering to design reflectors and antennas. Hyperbolas are used in physics and engineering to describe the behavior of waves and particles.In conclusion, conic sections are a fascinating topic in mathematics with diverse applications in various fields. Understanding the properties and equations of circles, ellipses, parabolas, and hyperbolas allows us to analyze and solve problems involving these curves. By studying conic sections, we gain valuable insights into the fundamental principles of geometry and their practical applications.。
轨道角动量算符的定义
轨道角动量算符的定义
在量子力学里，角动量算符(angular momentum operator)是一种算符，类比于经典的角动量。
在原子物理学涉及旋转对称性(rotational symmetry)的理论里，角动量算符占有中心的角色。
角动量，动量，与能量是物体运动的三个基本特性。
角动量促使在旋转方面的运动得以数量化。
在孤立系统里，如同能量和动量，角动量是守恒的。
在量子力学里，角动量算符的概念是必要的，因为角动量的计算实现于描述量子系统的波函数，而不是经典地实现于一点或一刚体。
在量子尺寸世界，分析的对象都是以波函数或量子幅来描述其概率性行为，而不是命定性(deterministic)行为。