计算机图形学基础chap4

Computer GraphicsShi-Min HuShi-Min HuTsinghua UniversityHistory of Ray Tracing(History of Ray Tracing (光线跟踪)•In 1980, Whitted proposed a ray tracing model, include light reflection and model include light reflection andtransmission effects. A Milestone of Computer Graphics.•Turner Whitted ，An improved illumination model for shaded display Communications of model for shaded display，Communications of the ACM, v.23 n.6, p.343-349, June 1980.•/study/home.htm http://www raytracing co uk/study/home htm•After receiving his PhD from NCSU in 1978,T Whitt d l ft f B ll L b d d d t Turner Whitted left for Bell Labs and proceeded to shake the CGI world with an algorithm that could ray-trace a scene in a reasonable amount of time. He only has14papers and Ray tracing is his first •He only has 14 papers, and Ray tracing is his first paper.•He was elected as member of National Academy of Engineering in 2003.of Engineering in2003Ray Tracing(光线跟踪)Ray Tracing(•Introduction of ray tracing•Ray intersection(光线求交)•shadows(阴影)p p•Transparence and specular reflection (透明和镜面反射)•textures(纹理)Introduction of ray tracing Introduction of ray tracing•Ray tracing–Because of its effectiveness, ray tracing is awidely used and very powerful rendering(drawing) techniqueWhy we see objects?–Why we see objects?•Light can be interpreted as a collection ofrays that begin at the light sources andrays that begin at the light sources andbounce around the objects in the scenes.•We see objects become rays finally come intoW bj b fi ll iour eyes.Basic Idea of ray tracingBasic Idea of ray tracing•Think of the frame buffer as a simple array of p,y g gpixels, with eye looking through it into thescenep,•For each pixel, what can we “see”?• a ray casting from the eye through the center of the pixel and out into the scene, its path is p,ptraced to see which object the ray hits firstg p •calculate the shading value of the pointby the Phong modelcontinue to trace the ray•continue to trace the rayin the scene to achievereflection refractionIntroduction of ray tracing Introduction of ray tracing•Features–Easy to incorporate interesting visual effects,Easy to incorporate interesting visual effectssuch as shadowing, reflection and refraction,since the path of a ray is traced through thei th th f i t d th h thscene–Besides geometric primitives(p,,),(such as spheres, cones, cubes),easy to work with a richerclass of objects includingclass of objects, includingpolygonal meshes, compoundobjectsRecursive Ray TracingRecursive Ray TracingIntersectColor( vBeginPoint, vDirection){Determine IntersectPoint;ete e te sect o t;Color = ambient color;for each lightColor += local shading term;Color+=local shading term;if(surface is reflective)color += reflect Coefficient *IntersectColor(IntersecPoint, Reflect Ray);IntersectColor(IntersecPoint Reflect Ray) else if ( surface is refractive)color += refract Coefficient *IntersectColor(IntersecPoint, Refract Ray);return color;}demos of ray tracingdemos of ray tracing•DENG Jia (计2 邓嘉)’s DemoPlay Video•Deng Jia, 2002-2006, Tsinghua University, No. 1 in GPA.No1in GPA•2006-, Princeton Univetrsity, published a paper on relief in ACM SIGGRAPH 2007•Deng Jia’s story in Media computingD Ji’t i M di ti (late)Ray intersection (光线求交) Ray intersection(•Ray tracingR R t ti–Ray Representation–Plane intersection–Triangle intersectionP l i t ti–Polygon intersectionp–Sphere intersection–Box intersectionRay representation Ray representation•Parametric representationP(t)=R +t *R –P(t) = R o + t * R d–where R o =(x o ,y o ,z o ) is the original point of the ray , R y z )is the direction the ray is going on usually d =(x d ,y d ,z d ) is the direction the ray is going on, usually the direction is normalizedt value determines the point the ray arrives at its value –t value determines the point the ray arrives at, its value is always larger than 0P(t)y gi i direction R d R ooriginPlane Intersection Plane Intersection•Plane DefinitionExplicit:P y z )n=(A B C)–Explicit: P o =(x o ,y o ,z o ), n=(A,B,C)–Implicit:H(P) = Ax+By+Cz+D = 0=n·P +D =0= n·P + D = 0•Point Plane Distance H(p) = d > 0–If n is normalized,the distance is d = H(P)normal P –But note that, d is signed distance HP o P'H(p) = d < 0Plane IntersectionPlane Intersection•Where does the ray intersect this plane?Given a plane with equation:•Given a plane with equation:n·P + D = 0;•Intersection means satisfy both equations: I t ti ti f b th tiP(t) = R o+ t * R dn·P(t)+D = 0;P()D0So we have()/(t = -(D+n·R o)/(n· R d)•Just need to verify if t>0P(t)Triangle IntersectionTriangle Intersection•In real-time graphics, triangle geometry (triangle mesh)is usualy stored and each (triangle mesh) is usualy stored, and eachtriangle is defined by three vertices.•There exists many different ray/triangle Th i t diff t/t i l intersection methods, main steps are:–First compute the intersection point betweenthe ray and the triangle’s plane–Thereafter, project the intersection pointonto the triangle’s plane–Decide whether or not the point is inside thetriangleTriangle Intersection Triangle Intersection•Barycentric coordinates:A point P on a triangle P is gi en b –A point P, on a triangle P 0P 1P 2, is given by the explicit formula:where are the barycentric 012P P P P αβγ=++(,,)αβγβy coordinates, which must satisfy–Note that the barycentric coordinates 0,,1,1αγαβγ≤≤++=P 0Note that the barycentric coordinates could be used for texture mapping, normal interpolation P normal interpolation, color interpolation, etc.P 1P 2Triangle Intersection Triangle Intersection•Since α + β + γ=1, we can write α =1−β −γ, We have:Set ray equation equal to barycentric equation:012(1)P P P P βγβγ=−−++•Set ray equation equal to barycentric equation:h i 012+(1)o d R tR P P P βγβγ=−−++⎛•Rearrange the terms, gives:010200( )d t R P P P P P R β⎞⎜⎟−−=−⎜⎟•The means the barycentric coordinate and distance γ⎜⎟⎝⎠yt can be found by solving this linear system of equationsTriangle Intersection Triangle Intersection•Denotingthe solution to the equation above is obtained by 10120200, E ,E P P P P S P R =−=−=−q y Cramer’s rule:12det(,,)S E E t ⎛⎞⎛⎞2121det(,,)det(,,)det(d S E d E E d E S β⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟Then check1(,,)γ⎝⎠⎝⎠0,1,1βγβγ≤≤+≤•Then check to determine whether or not the intersect point is inside the triangle gPolygon Intersection Polygon Intersection•Even though triangles are the most common rendering primitive a routine to compute the rendering primitive, a routine to compute the intersection between a ray and a polygon is useful.• A polygon of n vertices is defined by an ordered p yg y vertex list {v 0,v 1,…,v n-1}, where vertex v i forms an edge with for 0<=i <n 1and the polygon is an edge with for 0 <= i < n-1, and the polygon is closed by the edge from v n-1to v 0. The plane of h l i d d b the polygon is denoted by : 0p p p n x d π⋅+=Polygon IntersectionPolygon Intersection•We first compute theintersection between the ray intersection between the raybeen presented in previous slides.been presented in previous slides•we need to determine whetherd t d t i h thor not the intersection point P is inside the polygon.th l•This is done by projecting all vertices andy p j gP to one of the xy-,xz-,yz planes, as shown in the figurePolygon IntersectionPolygon Intersection•The question left is a 2D point-in-polygonproblem. Here we review one of the most bl H i f th tuseful algorithm –the crossing testg g•The crossing test is based on JordanC Th hi h th t i t i Curve Theorem,which says that a point is inside a polygon if a ray from this pointyg yin an arbitrary direction crosses anodd(奇数的) number of edges. This test is)number of edges This test isalso known as the even-odd test.Polygon IntersectionPolygon Intersection•The crossing test is illustrated in the right figure:figure:–Two black points inside polygon (cross oneedge)edge)–two black points outside polygon (onecrosses2edges and•does not usep p gpreprocessing(预处理)Polygon IntersectionPolygon Intersection•The test point P can also be thought of asbeing at the origin, and the edges may be b i t th i i d th d btested against the positive x-axis insteadg p–If the y-coordinates of the edge have thesame sign then the edge does not cross x axissame sign, then the edge does not cross x-axis –Else, compute the x-coordinate of theintersection between x-axis and the edge•If positive, the edge crosses x-axis, the number ofcrossing increases by 1•Else, the edge does not cross the x-axis•The other methodSphere Intersection Sphere Intersection•Now, let’s look at the intersection test between a ray and a sphere between a ray and a sphere•Mathematical solution–sphere defined as:•a center point , and a radius r ; the implicit formula for the sphere is :c P formula for the sphere is :To solve for the intersection between a ray ()0cf P P P r =−−=–To solve for the intersection between a ray and a sphere, simply replace P in the ray equationto yield:equation to yield:()0c P t P r −−=Sphere Intersection:Algebra method Sphere Intersection: Algebra method •The equation of last page is simplified as follows:()0c P t P r −−=0d c R tR P r+−=200()()d c d c R tR P R tR P r+−⋅+−=22000()2(())()()0d d d c c c t R R t R R P R P R P r ⋅+⋅−+−⋅−−=Since Rd is normalized:220002(())()()0d c c c t t R R P R P R P r +⋅−+−⋅−−=2=Rewrite as:20t tb c ++ew te as:2t b b c =−±−Solution is:Sphere IntersectionSphere Intersection•Previous algebra solution could be improved, e.g., observing thati d b i th t intersections behind the ray origin arey gnot neededA ti i d S l ti G t i •An optimized Solution: Geometric method–Easy reject (No intersection testing)Easy to check ray origin inside or outside the –Easy to check ray origin inside or outside thesphereE t h k hi h i t i l d t fSphere Intersection:Geometric Method Sphere Intersection: Geometric Method •Optimized SolutionWe first compute the vector from the ray l r –We first compute the vector from the ray origin to the center of the sphere:l P R =−r 0c PcrR oR dSphere Intersection Sphere Intersection•Optimized Solution compute the vector from the ray origin tol r –compute the vector from the ray origin to the center of the sphere:l P R =−r –Is ray origin inside/outside the sphere?I id th h 0c 22l r <r •Inside the sphere:•Outside the sphere:On the sphere:22l r >r 22l r =r •On the sphere:–If ray origin is on the sphere, be careful about degeneracy about degeneracySphere Intersection Sphere Intersection–Next, Find closest point to sphere center:d t l R =⋅r If origin outside &t <0no hit p •If origin outside & t P < 0 → no hit –the sphere is behind the ray origin and we can reject the intersection we can reject the intersectionPcrR o R d t PSphere Intersection Sphere Intersection–Next, find squared distance d from thesphere center to the closest point p p –If d>r no hit222pd l t =−r If d>r , no hit PPc drR oR d t PSphere Intersection Sphere Intersection–Find distance (t ’) from closest point (t P ) tocorrect intersection:’=r correct intersection: t 2= r 2-d 2–Then the solution will be•If origin outside sphere →t = t P -t ’ ()•If origin inside sphere →t = t + t ’P OR o R drtd t Pt ’Box IntersectionBox Intersection•Ray/Box intersection test are important ingraphics, since we often bound geometricgraphics since we often bound geometricobjects with boxes, which is called bounding box(包围盒).b()–With bounding box, whentesting a ray intersect witht ti i t t ithan object, we first test the rayintersect with the bounding box,intersect with the bounding boxif not intersected, then the raywill not intersect with the objectwill not intersect with the objectdefinitely.(early quiting)(提前退出)Box IntersectionBox Intersection•Here, we introduce a slab based method for box intersection test, proposed byf b i t ti t t d b Haines–A slab is simply two parallel planes, whichare grouped for faster computationare grouped for faster computation–A box is composed of 3 slabsin 2DBox Intersection Box Intersection•For each slab, intersecting with the ray, there is a minimum t value and i there is a minimum t value and maximum t value, these are called and (i=012)maxit min i t and . (i=0,1,2)•The next step is to compute:min min min min 012max(,,);t t t t =max max max max 012min(,,);t t t t =min max t t<•Now, the clever test: if then the ray intersects the boxmin max t t <min t max t–if ,then the ray intersects the box is the enter-point, is the exit point Otherwise no intersectionsBox Intersection Box Intersection•there are 2rays:1there are 2 rays:–with the box with the box it hasTh i ht i 0tmin maxt t<–The right ray misses it does not havei min t minmax•The idea is the same with min tmintt<The idea is the same with tBox Intersection Box Intersection•Woo’s Method for intersection between a d i li d b (ray and an axis-aligned box(和坐标轴平行)–ifi f i li d b–specific for axis-aligned box i i n1n 0Box IntersectionBox Intersection•Woo’s MethodFirst identity three candidate planes out of–First, identity three candidate planes out ofthe six planes. For each pair of parallelplanes, the back facing plane can be omittedplanes,the back–facing plane can be omittedfor furtherconsideration.–we computed the intersectionpmray and the planes.–Box IntersectionBox Intersection•Woo’s MethodUse the potential hit t value to compute the–Use the potential hit t-value to compute theintersection point–if the intersection point is located on the faceif the intersection point is located on the faceof the box, then it is a real hit•Slabs method vs Woo’s MethodSlabs method vs Woo’s Method–Comparable in performancey,•With above discussion of ray intersection, let’s ray tracingThe Simplest Ray Tracing Ray Casting The Simplest Ray Tracing：Ray Casting•For each pixel–Cast a ray from the eye, through the centerCast a ray from the eye through the centerof pixel, to the scene–For each object in the sceneFor each object in the scene•Find the intersection with the rayStore the closest point•Store the closest point–Calculate localshading term of theshading term of thepoint according tolight, materialg,and normalThe Simplest Ray Tracing: Ray Casting The Simplest Ray Tracing:Ray Casting•Shading results depend on surfacel li ht di ti li ht i t it normal, light direction, light intensity, view direction, material property and so on•Do not account for secondaryDo not account for secondaryray, so do not have shadow,reflection and refraction effectsSample Code of Ray casting Sample Code of Ray casting•for each pixel–cast a ray and find the intersection pointcast a ray and find the intersection point –if have intersectionl bi t•color = ambient•for each lightcolor+=shading from this lightcolor + shading from this light(depending on light property andmaterial property)•return colorreturn color–Elset b k d l•return background_colorAdd ShadowsAdd Shadows•For each pixelCast a ray and find the intersection point–Cast a ray and find the intersection point–color = ambient–for each lightf h li ht•if intersection point is not in shadow area of the light(evaluated by shadow rays)(evaluated by shadow rays)color += shading from this light–return colorreturn colorAdd ShadowsAdd Shadows•As illustrated in the figure below, casting ashadow ray from intersection point to the light, if shadow ray from intersection point to the light if there is an intersection, this point is in shadow •We only want to know whether there is anWe only want to know whether there is anintersection, not which one is closestAdd Ray reflection and refractionAdd Ray reflection and refraction•Ray tracing gives the ability to have objects ith i fl ti bj t ithwith mirror reflections or objects withrefractions.–The first step is to determine where a rayintersects the object.–The next step is to determine the direction the raywill travel when it reflects off the surface orrefracts through the object.–a new ray direction is calculated based on thea new ray direction is calculated based on theincoming ray and the surface normal.Ray Reflections Ray Reflections•The law of reflection:the angle of incidence =the angle of reflection–the angle of incidence = the angle of reflection –incoming ray, reflection ray and normal is in the same plane•The reflection ray is calculated as:h I N d R2()R I I N N=−⋅–where I,N and R are all unit vectorsRay Reflections Ray Reflections•As in the figure, reflection ray is symmetric with respect to the normal symmetric with respect to the normal from the view directionRay RefractionRay Refraction•When light travels from one transparent medium into another the direction of light can changeinto another, the direction of light can changebecause of the relative densities of the mediaRay RefractionRay Refraction•The law of refraction (also called Snell’s Law):–the ratio of the sin es of the angles of incidence and ofrefraction is a constant that depends on the media–The constant is called the relative refractive indexRay Refractions Ray Refractions•Snell’s law gives:N –M sin Өi sin sin i i T Tηθηθ=I Өi M ηT ηi N cos Өi 2222sin sin i i T T ηθηθ=T ӨT-N 2222(1cos )(1cos )i i T T ηθηθ−=−22N (1cos )cos T ηθθ−=The transmitted ray direction can now be calculated by :(T η∗(cos )i i T T T I N ηθηη=+∗。