Ray Casting. Outline. Outline in Code

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1 Foundaions of ompuer Graphics Online Lecure 10: Ray Tracing 2 Nus and ols amera Ray asing Ravi Ramamoorhi Ouline amera Ray asing (choose ray direcions) Ray-objec inersecions Ray-racing ransformed objecs Lighing calculaions Recursive ray racing Ouline in ode Image Rayrace (amera cam, Scene scene, in widh, in heigh) { Image image = new Image (widh, heigh) ; for (in i = 0 ; i < heigh ; i++) for (in j = 0 ; j < widh ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Inersecion hi = Inersec (ray, scene) ; image[i][j] = Findolor (hi) ; reurn image ; Virual Viewpoin Ray asing Virual Screen Objecs Ray Muliple misses inersecs inersecions: all objec: objecs: shade Pixel Use using colored closes color, black one lighs, (as does maerials OpenGL) Finding Ray Direcion Goal is o find ray direcion for given pixel i and j Many ways o approach problem Objecs in world coord, find dirn of each ray (we do his) amera in canonical frame, ransform objecs (OpenGL) asic idea Ray has origin (camera cener) and direcion Find direcion given camera params and i and j Similar o glulook derivaion glulook(eyex, eyey, eyez, cenerx, cenery, cenerz, upx, upy, upz) amera a eye, looking a cener, wih up direcion being up Eye Up vecor amera params as in glulook Lookfrom[3], Look[3], up[3], fov From earlier lecure on deriving glulook ener 1

2 onsrucing a coordinae frame? We wan o associae w wih a, and v wih b u a and b are neiher orhogonal nor uni norm nd we also need o find u w = a a u = b w b w v = w u From basic mah lecure - Vecors: Orhonormal asis Frames onsrucing a coordinae frame w = a a u = b w b w We wan o posiion camera a origin, looking down Z dirn Hence, vecor a is given by eye cener The vecor b is simply he up vecor Eye v = w u Up vecor ener anonical viewing geomery anonical viewing geomery -w αu βv -w αu βv α = an fovx j (widh / 2) 2 widh / 2 β = an fovy 2 (heigh / 2) i heigh / 2 anonical viewing geomery Foundaions of ompuer Graphics -w αu βv αu + βv w ray = eye + αu + βv w Online Lecure 10: Ray Tracing 2 Nus and ols Ray-Objec Inersecions α = an fovx j (widh / 2) 2 widh / 2 β = an fovy 2 (heigh / 2) i heigh / 2 Ravi Ramamoorhi 2

3 Ouline amera Ray asing (choosing ray direcions) Ray-objec inersecions Ray-racing ransformed objecs Lighing calculaions Recursive ray racing Ouline in ode Image Rayrace (amera cam, Scene scene, in widh, in heigh) { Image image = new Image (widh, heigh) ; for (in i = 0 ; i < heigh ; i++) for (in j = 0 ; j < widh ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Inersecion hi = Inersec (ray, scene) ; image[i][j] = Findolor (hi) ; reurn image ; Ray-Sphere Inersecion sphere ( P ) i ( P ) r 2 = 0 Ray-Sphere Inersecion sphere ( P ) i ( P ) r 2 = 0 Subsiue P 0 Ray-Sphere Inersecion sphere ( P ) i ( P ) r 2 = 0 Subsiue sphere ( P 0 ) i ( P 0 ) r 2 = 0 Simplify Ray-Sphere Inersecion sphere ( P ) i ( P ) r 2 = 0 Subsiue sphere ( P 0 ) i ( P 0 ) r 2 = 0 Simplify 2 ( P 1 i P 1 ) + 2 P 1 i ( P 0 ) + ( P 0 ) i ( P 0 ) r 2 = 0 3

4 Ray-Sphere Inersecion 2 ( P 1 i P 1 ) + 2 P 1 i ( P 0 ) + ( P 0 ) i ( P 0 ) r 2 = 0 Solve quadraic equaions for 2 real posiive roos: pick smaller roo oh roos same: angen o sphere One posiive, one negaive roo: ray origin inside sphere (pick + roo) omplex roos: no inersecion (check discriminan of equaion firs) Ray-Sphere Inersecion Inersecion poin: Normal (for sphere, his is same as coordinaes in sphere frame of reference, useful oher asks) P normal = P Ray-Triangle Inersecion One approach: Ray-Plane inersecion, hen check if inside riangle Plane equaion: Ray-Triangle Inersecion One approach: Ray-Plane inersecion, hen check if inside riangle Plane equaion: n = ( ) ( ) ( ) ( ) Ray-Triangle Inersecion One approach: Ray-Plane inersecion, hen check if inside riangle Plane equaion: plane P i n i n = 0 n = ( ) ( ) ( ) ( ) Ray-Triangle Inersecion One approach: Ray-Plane inersecion, hen check if inside riangle Plane equaion: plane P i n i n = 0 ombine wih ray equaion ( P 0 ) i n = i n n = ( ) ( ) ( ) ( ) = i n P 0 i n P 1 i n 4

5 Ray inside Triangle Once inersec wih plane, need o find if in riangle Many possibiliies for riangles, general polygons We find paramerically [barycenric coordinaes]. lso useful for oher applicaions (exure mapping) α β P γ P = α + β + γ α 0, β 0,γ 0 α + β + γ = 1 Ray inside Triangle α β P P = α + β + γ α 0, β 0,γ 0 α + β + γ = 1 γ P = β( ) + γ ( ) 0 β 1, 0 γ 1 β + γ 1 Oher primiives Much early work in ray racing focused on ray-primiive inersecion ess ones, cylinders, ellipsoids oxes (especially useful for bounding boxes) General planar polygons Many more Ray Scene Inersecion Inersecion (ray, scene) { mindis = infiniy; hiobjec = NULL ; For each objec in scene { // Find closes inersecion; es all objecs = Inersec (ray, objec) ; if ( > 0 && < mindis) // closer han previous closes objec mindis = ; hiobjec = objec ; reurn InersecionInfo(mindis, hiobjec) ; // may already be in Inersec() Ouline amera Ray asing (choosing ray direcions) Ray-objec inersecions Ray-racing ransformed objecs Lighing calculaions Recursive ray racing Ray-Tracing Transformed Objecs We have an opimized ray-sphere es u we wan o ray race an ellipsoid Soluion: Ellipsoid ransforms sphere pply inverse ransform o ray, use ray-sphere llows for insancing (raffic jam of cars) Same idea for oher primiives 5

6 Transformed Objecs onsider a general 4x4 ransform M (marix sacks) pply inverse ransform M -1 o ray Locaions sored and ransform in homogeneous coordinaes Vecors (ray direcions) have homogeneous coordinae se o 0 [so here is no acion because of ranslaions] Do sandard ray-surface inersecion as modified Transform inersecion back o acual coordinaes Inersecion poin p ransforms as Mp Normals n ransform as M - n. Do all his before lighing Foundaions of ompuer Graphics Online Lecure 10: Ray Tracing 2 Nus and ols Lighing alculaions Ravi Ramamoorhi Ouline amera Ray asing (choosing ray direcions) Ray-objec inersecions Ray-racing ransformed objecs Lighing calculaions Recursive ray racing Ouline in ode Image Rayrace (amera cam, Scene scene, in widh, in heigh) { Image image = new Image (widh, heigh) ; for (in i = 0 ; i < heigh ; i++) for (in j = 0 ; j < widh ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Inersecion hi = Inersec (ray, scene) ; image[i][j] = Findolor (hi) ; reurn image ; Shadows Ligh Source Shadows: Numerical Issues Numerical inaccuracy may cause inersecion o be below surface (effec exaggeraed in figure) ausing surface o incorrecly shadow iself Move a lile owards ligh before shooing shadow ray Virual Viewpoin Virual Screen Objecs Shadow ray o o ligh is is unblocked: objec visible in shadow 6

7 Similar o OpenGL Lighing Model Lighing model parameers (global) mbien r g b L enuaion cons linear quadraic L = 0 cons + lin * d + quad * d 2 Per ligh model parameers Direcional ligh (direcion, RG parameers) Poin ligh (locaion, RG parameers) Some differences from HW 2 synax Maerial Model Diffuse reflecance (r g b) Specular reflecance (r g b) Shininess s Emission (r g b) ll as in OpenGL n Shading Model I = K a + K e + V i L i (K d max (l i i n,0) + K s (max(h i i n,0)) s ) i=1 Global ambien erm, emission from maerial For each ligh, diffuse specular erms Noe visibiliy/shadowing for each ligh (no in OpenGL) Evaluaed per pixel per ligh (no per verex) Foundaions of ompuer Graphics Online Lecure 10: Ray Tracing 2 Nus and ols Recursive Ray Tracing Ravi Ramamoorhi Ouline amera Ray asing (choosing ray direcions) Ray-objec inersecions Ray-racing ransformed objecs Lighing calculaions Recursive ray racing Mirror Reflecions/Refracions Virual Viewpoin Virual Screen Generae refleced ray in mirror direcion, Ge reflecions and refracions of objecs Objecs 7

8 asic idea For each pixel Trace Primary Eye Ray, find inersecion Trace Secondary Shadow Ray(s) o all ligh(s) olor = Visible? Illuminaion Model : 0 ; Trace Refleced Ray olor += refleciviy * olor of refleced ray Recursive Shading Model n I = K a + K e + V i L i (K d max (l i i n,0) + K s (max(h i i n,0)) s ) + K s I R + K T I T i=1 Highlighed erms are recursive speculariies [mirror reflecions] and ransmission (laer is exra) Trace secondary rays for mirror reflecions and refracions, include conribuion in lighing model Geolor calls RayTrace recursively (he I values in equaion above of secondary rays are obained by recursive calls) Problems wih Recursion Reflecion rays may be raced forever Generally, se maximum recursion deph Same for ransmied rays (ake refracion ino accoun) Some basic add ons rea ligh sources and sof shadows: break ino grid of n x n poin lighs Use jiering: Randomize direcion of shadow ray wihin small box for given ligh source direcion Jiering also useful for anialiasing shadows when shooing primary rays More complex reflecance models Simply updae shading model u a presen, we can handle only mirror global illuminaion calculaions 8