Repetition of TDA361. Misc. GEOMETRY Summary. Lecture 2: Transforms. Homogeneous notation. Ulf Assarsson

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1 Repetton of TDA361 Ulf Assarsson Msc Eng translaton: All your answers on exam must be wrtten n Englsh Tll alla lärare på masternvån, Undervsnngen på Chalmers masterprogram sker på engelska. Vcerektor Sven Engström klarlägger att all examnaton också sker på engelska dvs tentamenssvaren skall vara på engelska. Syftet är att studenten skall träna att kommuncera på engelska såväl muntlgt som skrftlgt olka stuatoner. För skrftlga tentamna gäller t ex att annat språk än engelska nte skall beaktas vd rättnngen. Som tdgare är ordböcker tllåtna hjälpmedel vd tentamna förutom vd tentamen språk. Var vänlg meddela detta tll studenterna på Dna kurser nför tentamensperoden oktober. Lecture 1: Real-tme Renderng The Graphcs Renderng Ppelne Chapter 2 n the book The ppelne s the engne that creates mages from 3D scenes Three conceptual stages of the ppelne: Applcaton (executed on the CPU) Geometry Rasterer GEOMETRY Summary model space world space world space Applcaton Geometry Rasterer camera space nput Applcaton Geometry Rasterer 3D scene + how to locate the Image output compute lghtng projecton mage space bottleneck, RTR: ch 1.1 ModelVewMtx = Model to Vew Matrx clp map to screen Lecture 2: Transforms Homogeneous notaton Quaternons Know what they are good for. Not knowng all mathematcal rules. Projectons 3. Vectors and Transforms rotatonsdelen Homogeneous notaton A pont: p px py p 1 Translaton becomes: 1 tx px px tx Translatonsdelen 1 t y py py t y 1 t p p t T( t) A vector (drecton): d d x d y d Translaton of vector: Td d Also allows for projectons (later) T T 1

2 3. Vectors and Transforms M Change of Frames M model-to-world : a y ax ay a b b b x y c c c x y model-to-world ox o y o 1 world space x (,5,) b o model space c 3. Vectors and Transforms Projectons Orthogonal (parallel) and Pertve E.g.: p world = M mw p model = M mw (,5,) T = 5 b (+ o) 3. Vectors and Transforms Orthogonal projecton Smple, just skp one coordnate Say, we re lookng along the -axs Then drop, and render M ortho M ortho px px py py p Rasteraton, Depth Sortng and Cullng: DDA Algorthm Dgtal Dfferental Analyer DDA was a mechancal devce for numercal soluton of dfferental equatons Lne y=kx+ m satsfes dfferental equaton dy/dx = k = Dy/Dx = y 2 -y 1 /x 2 -x 1 Along scan lne Dx = 1 y=y1; For(x=x1; x<=x2,x++) { wrte_pxel(x, round(y), lne_color) y+=k; } 6. Rasteraton, Depth Sortng and Cullng: Usng Symmetry Use for 1 k For k > 1, swap role of x and y For each y, plot closest x 6. Rasteraton, Depth Sortng and Cullng: Very Important! The problem wth DDA s that t uses floats whch was slow n the old days Bresenhams algorthm only uses ntegers You do not need to know Bresenham s algorthm by heart. It s enough to understand the followng 2 sldes. 2

3 Bresenham s lne drawng algorthm The lne s drawn between two ponts (x, y ) and (x 1, y 1 ) ( y Slope 1 y) k (y = kx + m) ( x1 x) Each tme we step 1 n x-drecton, we should ncrement y wth k. Otherwse the error n y ncreases wth k. If the error surpasses.5, the lne has become closer to the next y- value, so we add 1 to y smultaneously decreasng the error by 1 functon lne(x, x1, y, y1) nt deltax := abs(x1 - x) nt deltay := abs(y1 - y) real error := real deltaerr := deltay / deltax nt y := y for x from x to x1 plot(x,y) error := error + deltaerr f error.5 y := y + 1 error := error - 1. Ths s the reason that DDA needs floats. So Bresenham s soluton s to multply all correspondng numbers wth (x 1 -x ) to get ntegers. See also Ulf Assarsson 26 Bresenham s lne drawng algorthm Now, convert algorthm to only usng nteger computatons The trck we use s to multply all the fractonal numbers above by (x 1, x ), whch enables us to express them as ntegers. The only problem remanng s the constant.5 to deal wth ths, we multply both sdes of the nequalty by 2 Old float verson: functon lne(x, x1, y, y1) nt deltax := abs(x1 - x) nt deltay := abs(y1 - y) real error := real deltaerr := deltay / deltax nt y := y for x from x to x1 plot(x,y) error := error + deltaerr f error.5 y := y + 1 error := error - 1. New nteger verson: functon lne(x, x1, y, y1) nt deltax := abs(x1 - x) nt deltay := abs(y1 - y) real error := real deltaerr := deltay nt y := y for x from x to x1 plot(x,y) error := error + deltaerr f 2*error deltax y := y + 1 error := error - deltax Bresenhams alg. Ulf Assarsson Rasteraton, Depth Sortng and Cullng: Panter s Algorthm Render polygons a back to front order so that polygons behnd others are smply panted over B behnd A as seen by vewer Requres orderng of polygons frst O(n log n) calculaton for orderng Not every polygon s ether n front or behnd all other polygons Fll B then A I.e., : Sort all trangles and render them back-to-front. 6. Rasteraton, Depth Sortng and Cullng: -Buffer Algorthm Use a buffer called the or depth buffer to store the depth of the closest object at each pxel found so far As we render each polygon, compare the depth of each pxel to depth n buffer If less, place shade of pxel n color buffer and update buffer Lekton 3: OpenGL Uses OpenGL (or DrectX) Wll not ask about syntax. Know how to use. E.g. how to acheve Transparency Fog(start, stop, lnear/exp/exp-squared) Specfy a materal, a trangle 3. OpenGL: Specfyng vertces and polygons OpenGL s a state machne. Commands typcally change the current state Multple callng formats for the commands: glvertex{234}{sfd}( coords ) glbegn()/glend(). (Slow) glbegn(gl_triangle) glvertex3f(,,) glvertex3f(,1,); glvertex3f(1,1,); glend(); Optonal: Can fy for nstance glcolor3f(r,g,b), gltexcoord2f(s,t), glnormal3f(x,y,) - typcally per vertex or per prmtve. Dsplay lsts are created wth surroundng glnewlst() and glendlst(). (Fast) Idea: Objects are nt dlst=glgenlsts(1); // generate one dsplay lst number glnewlst(dlst,gl_compile); // ndcates start of dsplay lst stored nternally n // or any object fcaton the most effcent glendlst(); // ndcates end of dsplay lst format Dsplay lsts are then drawn wth: glcalllst(dlst); Vertex Arrays (Fast): gltexcoordponter( se type strde *ponter ) //Defnes an array of texture coordnates glcolorponter( strde *ponter ) //Defnes an array of colors se type glindexponter( type strde *ponter ) //Defnes an array of color ndces) glnormalponter( type strde *ponter ) //Defnes an array of normals glvertexponter( se type strde *ponter ); //Defnes an array of vertces glenableclentstate( array gldsableclentstate( array wth array set to ) ), or,,,, glarrayelement(nt ); // Transfers the th element of every enabled array to OpenGL gldrawarrays mode; frst; count // Renders multple prmtves from the enabled arrays // Same as gldrawarrays, but takes array gldrawelements(mode, count type *ndces ) elements ponted to by ndces. glinterleavedarrays format; strde; ponter // One array wth vertces, normals, textures and/colors nterleaved Ulf Assarsson 23 3

4 3. OpenGL: Matrx and Stack Operatons Matrx Operatons: glrotate(),gltranslate(), glscale(), GLU Helper functons: glupertve(), glulookat, Ulf Assarsson OpenGL: Lghtng and Colors glcolor4f(r,g,b,a), glcolor3f(r,g,b) Used when lghtng s dsabled: Dsable wth gldsable( GL_LIGHTING ); Could be changed for nstance per vertex or per object. Can also be fed wth glcolorponter() as descrbed prevously glmateralfv() Used when lghtng s enabled. Enable wth glenable( GL_LIGHTING ); Must also enable lghts: glenable( GL_LIGHTn ); Example: glmateralfv(gl_front_and_back, GL_AMBIENT, float rgba[4]) glmateralfv(gl_front_and_back, GL_DIFFUSE, float rgba[4]) glmateralfv(gl_front_and_back, GL_SPECULAR, float rgba[4]) glmateralfv(gl_front_and_back, GL_EMISSION, float rgba[4]) glmateralf(gl_front_and_back, GL_SHININESS, 3) Ulf Assarsson OpenGL: Buffers Frame buffer Back/front/left/rght gldrawbuffers() Alpha channel: glalphafunc() Depth buffer (-buffer) For correct depth sortng Instead of BSP-algorthm, panters algorthm Stencl buffer E.g. Shadow volumes and maskng renderng to non-rectangular regons, Accumulaton buffer Lecture 4.1: Shadng Lghtng =amb+dff++emsson + = + Know how to compute components. Also, Blnns and Phongs hghlght model Ulf Assarsson Shadng: Lghtng Lght: Ambent (r,g,b,a) Dffuse (r,g,b,a) Specular (r,g,b,a) Materal: Ambent (r,g,b,a) Dffuse (r,g,b,a) Specular (r,g,b,a) Emsson (r,g,b,a) = självlysande färg Tomas Added Akenne-Mőller by Ulf Assarsson, Shadng: Lghtng =amb+dff++emsson I.e.: =amb+dff++emsson amb dff m amb ( nl) m Phong s reflecton model: Blnn s reflecton model: s dff amb s emsson m emsson dff msh max(,( r v) ) msh max(,( hn) ) m m s s 4

5 4. Shadng: Ambent component: amb Ad-hoc tres to account for lght comng from other surfaces Just add a constant color: amb m amb s amb.e., ( r, g, b, a ) = (m r, m g, m b, m a ) (l r, l g, l b, l a ) 4. Shadng: Dffuse component : dff =amb+dff++emsson Dffuse s Lambert s law: dff nl cos Photons are scattered equally n all drectons ( nl) m s dff dff dff gllghtmodelfv(gl_light_model_ambient, global_ambent) Modfed Tomas Akenne-Mőller by Ulf Assarsson, Shadng: Lghtng Specular component : 4. Shadng: Specular component: Phong Phong ular hghlght model Reflect l around n: r r l 2(n l)n n l nl Dffuse s dull (left) Specular: smulates a hghlght ( r v) (cos ) m sh m sh ( n l) n -l msh max(,( r v) ) m s Next: Blnns hghlght formula: (n. h) m 4. Shadng: Halfway Vector Blnn proposed replacng v r by n h where h = (l+v)/ l + v (l+v)/2 s halfway between l and v If n, l, and v are coplanar: y /2 Must then adjust exponent so that (n h) e (r v) e (e 4e) msh max(,( hn) ) m s 4. Shadng: Shadng Three common types of shadng: Flat, Goraud, and Phong In standard Gouraud shadng the lghtng s computed per trangle vertex and for each pxel, the color s nterpolated from the colors at the vertces. In Phong Shadng the lghtng s not per vertex. Instead the normal s nterpolated per pxel from the normals defned at the vertces and full lghtng s computed per pxel usng ths normal. Ths s of course more expensve but looks better. Flat Gouraud Phong 5

6 4. Shadng: Transparency and alpha Transparency Very smple n real-tme contexts The tool: alpha blendng (mx two colors) Alpha (a) s another component n the frame buffer, or on trangle Represents the opacty 1. s totally opaque. s totally transparent The over operator: o c ac ( 1a) c Rendered object s d 4. Shadng: Transparency Need to sort the transparent objects Frst, render all non-transparent trangles as usual. Then, sort all transparent trangles and render back-to-front wth blendng enabled. (and usng standard depth test) Partly to avod problems wth the depth test and partly because the blendng operaton s order dependent. Leture 4.2: Samplng, fltrerng, and AA Why care? When does t occur? In pxels, tme, texture, etc Nyqust Flters Supersamplng schemes Jttered samplng Adaptve samplng n ray tracng Flterng FILTERING: For magnfcaton: Nearest or Lnear (box vs Tent flter) For mnfcaton: Blnear usng mpmappng Trlnear usng mpmappng Ansotropc some mpmap lookups along lne Ulf of Assarsson ansotropy 24 Interpolaton Magnfcaton Blnear flterng usng Mpmappng Mnfcaton 6

7 7. Texturng: Mpmappng Image pyramd Half wdth and heght when gong upwards Average over 4 parent texels to form chld texel Dependng on amount of mnfcaton, determne whch mage to fetch from Compute d frst, gves two mages Blnear nterpolaton n each v d u 7. Texturng: Mpmappng Interpolate between those blnear values Gves trlnear nterpolaton Level n+1 (u,v,d ) d Level n v u Constant tme flterng: 8 texel accesses Ansotropc texture flterng 7. Texturng: 5. Texturng Most mportant: Texturng, envronment mappng Bump mappng 3D-textures, Partcle systems Sprtes and bllboards 7. Texturng: Envronment mappng Assumes the envronment s nfntely far away Sphere mappng For detals, see OH Cube mappng s the norm nowadays Advantages: no sngulartes as n sphere map Much less dstorton Gves better result Not dependent on a vew poston Modfed by Ulf Assarsson Texturng: Cube mappng eye y x n Smple math: compute reflecton vector, r Largest abs-value of component, determnes whch cube face. Example: r=(5,-1,2) gves POS_X face Dvde r by abs(5) gves (u,v)=(-1/5,2/5) If your hardware has ths feature, then t does all the work 7

8 7. Texturng: Bump mappng by Blnn n 1978 Inexpensve way of smulatng wrnkles and bumps on geometry Too expensve to model these geometrcally Instead let a texture modfy the normal at each pxel, and then use ths normal to compute lghtng per pxel 7. Texturng: 3D Textures 3D textures: Feasble on modern hardware as well Texture flterng s no longer trlnear Rather quadlnear (lnear nterpolaton 4 tmes) Enables new possbltes Can store lght n a room, for example + = geometry Bump map Bump mapped geometry Stores heghts: can derve normals 7. Texturng: Sprtes Sprtes (=älvor) was a technque on older home computers, e.g. VIC64. As opposed to bllboards sprtes does not use the frame buffer. They are rastered drectly to the screen usng a al chp. A al btregster also marked colldng sprtes. 7. Texturng: 2D mages used n 3D envronments Common for trees, explosons, clouds, lens flares Bllboards 7. Texturng: Bllboards 7. Texturng: Fx correct transparency by blendng AND usng alphatest glenable(gl_alpha_test); glalphafunc(gl_greater,.1); Bllboards Color Buffer Depth Buffer Rotate them towards vewer Ether by rotaton matrx or by orthographc projecton Threshold >. If alpha value n texture s lower than ths threshold value, the pxel s not rendered to. I.e., nether frame buffer nor -buffer s updated. Whch s what we want to acheve. E.g. here: Wth blendng Wth alpha test 8

9 7. Texturng: (Also called Impostors) n axal bllboardng The rotaton axs s fxed and dsregardng the vew poston Lecture 6.1: Intersecton Tests 4 technques: Analythc Geometrc e.g. ray vs box (3 slabs) SAT Test: 1. axes ortho to sde of A, 2. axes ortho to sde of B 3. crossprod of edges Dynamc tests Know how they work E.g., descrbe an algorthm for ntersecton between a ray and a polygon. Know equatons for ray, sphere, cylnder, plane Analytcal: Ray/plane ntersecton Ray: r(t)=o+td Plane formula: n p + d = Replace p by r(t) and solve for t: n (o+td) + d = n o+tn d + d = t = (-d -n o) / (n d) o d n Tomas Akenne-Mőller 23 Analytcal: Ray/sphere test Sphere center: c, and radus r Ray: r(t)=o+td Sphere formula: p-c =r Replace p by r(t): r(t)-c =r ( r( t) c) ( r( t) c) r 2 ( o td c) ( o td c) r ( dd) t t 2(( ο c) d) t ( ο c) ( ο c) r 2 2(( ο c) d) t ( ο c) ( ο c) r Ths s a standard quadratc equaton. Solve for t. o 2 d r c d 1 Tomas Akenne-Mőller 23 Lecture 6.2: Spatal Data Structures and Speed-Up Technques Speed-up technques Cullng Backface Vew frustum (herarchcal) Portal Occluson Cullng Detal Levels-of-detal How to construct and use the spatal data structures BVH, BSP-trees (polygon algned + axs algned) Lecture 6.2: Spatal data structures BVH, Grds (recursve/herarchcal), octrees/quadtree, BVH sphercal / OBB / AABB, BSP (polygon / axs algned, kdtrees) How to use, what they look lke, how to construct (f mentoned n the course) 9

10 Ray Tracng Summary of the Ray tracngalgorthm: Image plane trace() lght shade() trace() trace() shade() Pont s n shadow man()-calls trace() for each pxel trace(): should return color of closest ht pont along ray. 1. calls fndclosestintersecton() 2. If any object ntersected call shade(). Shade(): should compute color at ht pont 1. For each lght source, shoot shadow ray to determne f lght source s vsble If not n shadow, compute dffuse + ular contrbuton. 2. Compute ambent contrbuton 3. Call trace() recursvely for the reflecton- and refracton ray. Lecture 7-8: Ray tracng Adaptve Super Samplng Jtterng How to stop recurson? (Transmsson) Speedup technques Spatal data structures Optmatons for BVHs skpponter tree BVH-traversal (You do not need to learn the ray traversal algorthms for Grds nor AA-BSP trees) Shadow cache Materal (Fresnel: metall, delectrcs) Constructve Sold Geometry 9. Curves and Surfaces: Objectves Introduce the types of curves Interpolatng Blendng polynomals for nterpolaton of 4 control ponts (ft curve to 4 control ponts) Hermte ft curve to 2 control ponts + 2 dervatves (tangents) Beer 2 nterpolatng control ponts + 2 ntermedate ponts to defne the tangents B-splne To get C 2 contnuty NURBS Dfferent weghts of the control ponts and The control ponts can be at non-unform ntervalls p p 1 9. Curves and Surfaces: p 2 If we examne the cubc B-splne from the p 3 Splnes and Bass pertve of each control (data) pont, each nteror pont contrbutes (through the blendng functons) to four segments We can rewrte p(u) n terms of the data ponts as p( u) B ( u) p defnng the bass functons {B (u)} 9. Curves and Surfaces: B-Splnes p p1 p 2 p 3 p 4 p5 These are our control ponts, p - p 8, to whch we want to approxmate a curve p 6 p 7 p 8 SUMMARY p p 1 p 2 p 3 p 4 B-Splnes In each pont p(u) of the curve, for a gven u, the pont s defned as a weghted sum of the closest 4 surroundng ponts. Below are shown the weghts for each pont along u=1 1% Blendfuncton B 1(u) for pont p 1 p 5 p 6 p 7 p 8 1% u= Illustraton of how the control ponts are evenly (unformly) dstrbuted along the parametersaton u of the curve p(u). In each pont p(u) of the curve, for a gven u, the pont s defned as a weghted sum of the closest 4 surroundng ponts. Below are shown the weghts for each pont along u=1 p p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 u u u The weght functon (blend functon) B p (u) for a pont p can thus be wrtten as a translaton of a bass functon B(t). B p (u) = B(u-) B(t): 1% t Our complete B-splne curve p(u) can thus be wrtten as: p( u) B ( u) p 1

11 9. Curves and Surfaces: NURBS NURBS s smlar to B-Splnes except that: 1. The control ponts can have dfferent weghts, w, (hegher weght makes the curve go closer to that control pont) 2. The control ponts do not have to be at unform dstances (u=,1,2,3...) along the parametersaton u. E.g.: u=,.5,.9, 4, 14, NURBS = Non-Unform Ratonal B-Splnes The NURBS-curve s thus defned as: 9. Curves and Surfaces: NURBS Allowng control ponts at non-unform dstances means that the bass functons B p () are beng streched and non-unformly located. E.g.: Dvson wth the sum of the weghts, to make the combned weghts sum up to 1, at each poston along the curve. Otherwse, a translaton of the curve s ntroduced (whch can be fun but normally not desred) Each curve B p () should of course look smooth and C 2 contnuous. But t s not so easy to draw smoothly by hand... (The sum of the weghts are stll =1 due to the dvson n u Lecture 1.1: Collson Detecton 3 types of algorthms: Wth rays Fast but not exact Wth BVH Pseudo code s smple to derve Slower but exact For lots of objects. why? Course prunng of obvously non-colldng objects Sweep-and-prune 1.2 Vertex and Fragment Shaders: Vertex- and Fragment shaders Understand the followng shaders Eally the frst two 1. Vertex and Fragment Shaders: Wave Moton Vertex Shader 1. Vertex and Fragment Shaders: Partcle System unform float tme; unform float xs, s; vod man() { float s; s = *sn(xs*tme)*sn(s*tme); gl_vertex.y = s*gl_vertex.y; gl_poston = gl_modelvewprojectonmatrx*gl_vertex; } unform vec3 nt_vel; unform float g, m, t; vod man() { vec3 object_pos; object_pos.x = gl_vertex.x + vel.x*t; object_pos.y = gl_vertex.y + vel.y*t - g/(2.*m)*t*t; object_pos. = gl_vertex. + vel.*t; gl_poston = gl_modelvewprojectonmatrx* vec4(object_pos,1); } 11

12 1. Vertex and Fragment Shaders: Vertex Shader for per Fragment Lghtng /* vertex shader for per-fragment Phong shadng */ varyng vec3 normale; varyng vec4 postone; vod man() { normale = gl_normalmatrxmatrx*gl_normal; postone = gl_modelvewmatrx*gl_vertex; gl_poston = gl_modelvewprojectonmatrx*gl_vertex; } 1. Vertex and Fragment Shaders: Fragment Shader for per Fragment Lghtng varyng vec3 normale; varyng vec4 postone; vod man() { vec3 norm = normale(normale); vec3 lghtv = normale(gl_lghtsource[].poston-postone.xy); vec3 vewv = normale(postone); vec3 halfv = normale(lghtv + vewv); vec4 dffuse = max(, dot(lghtv, vewv)) *gl_frontmateral.dffuse*gl_lghtsource[].dffuse; vec4 ambent = gl_frontmateral.ambent*gl_lghtsource[].ambent; 1. Vertex and Fragment Shaders: Fragment Shader for per Fragment Lghtng nt f; f(dot(lghtv, vewv)>.) f =1.); else f =.; vec3 ular = f*pow(max(, dot(norm, halfv), gl_frontmateral.shnness) *gl_frontmateral.ular*gl_lghtsource[].ular); vec3 color = vec3(ambent + dffuse + ular); gl_fragcolor = vec4(color, 1.); } Lecture 11: Shadows + Reflecton Pont lght / Area lght Three ways of thnkng about shadows The bass for dfferent algorthms. Shadow mappng Be able to descrbe the algorthm Shadow volumes Be able to descrbe the algorthm Stencl buffer, 4-pass algorthm, Z-pass, Z-fal, Creatng quads from the slhouette edges as seen from the lght source, etc Pros and cons of shadow volumes vs shadow maps Planar reflectons how to do. Why not usng envronment mappng? Ways of thnkng about shadows As separate objects (lke Peter Pan's shadow) Ths corresponds to planar shadows As volumes of space that are dark Ths corresponds to shadow volumes As places not seen from a lght source lookng at the scene. Ths corresponds to shadow maps Note that we already "have shadows" for objects facng away from lght Usng the Shadow Map When scene s vewed, check vewed locaton n lght's shadow buffer If pont's depth s (epslon) greater than shadow depth, object s n shadow. shadow depth map For each pxel, compare dstance to lght wth the depth stored n the shadow map 1. Rendera shadow depth map from lght source. 2. Render from eye. For each generated pxel, transform (warp) the x,y,-coordnate to lght space and compare depth wth the stored depth value n the shadow map. If greater pont s n shadow Else pont s n lght Problem: bas / offset neededtomas Akenne-Mőller 22 12

13 Shadow volumes Shadow volume concept Create volumes of space n shadow from each polygon n lght. Each trangle creates 3 projectng quads Usng the Volume To test a pont, count the number of polygons between t and the eye. If we look through more frontfacng than backfacng polygons, then n shadow. backfacng frontfacng How to mplement shadow volumes wth stencl buffer (Z-pass) A four pass process [Hedmann91]: 1st Pass: render the scene wth just ambent lghtng. Turn off updatng Z-buffer and wrtng to color buffer (.e. Z-compare, draw to stencl only). 2nd pass: render front facng shadow volume polygons to stencl buffer, ncrementng count. 3rd pass: render backfacng shadow volume polygons to stencl, decrementng. 4th pass: render dffuse and ular where stencl buffer s. Compared to Z-pass: a=#shadow volumes a pont s located wthn. Invert -test a s ndependent of n whch drecton we count. Choose pont at nfnty. That pont s always Invert stencl nc/dec outsde shadow, due to the capng of the shadow I.e., count to nfnty nstead of from eye. volumes 13

14 Shadow maps vs shadow volumes Shadow Volumes Good: Anythng can shadow anythng, ncludng selfshadowng, and the shadows are sharp. Bad: 3 or 4 passes, shadow polygons must be generated and rendered lots of polygons & fll, Mergng Volumes Edge shared by two polygons facng the lght creates front and backfacng quad. Shadow Maps Good: Anythng to anythng, constant cost regardless of complexty, map can sometmes be reused. Bad: Frustum lmted. Jagged shadows f res too low, basng headaches. Ths nteror edge makes two quads, whch cancel out Slhouette Edges From the lght s vew, caster nteror edges do not contrbute to the shadow volume. Planar reflectons Assume plane s = Then apply glscalef(1,1,-1); Effect: Fndng the slhouette edge gets rd of many useless shadow volume polygons. Planar reflectons Backfacng becomes front facng! Lghts should be reflected as well Need to clp (usng stencl buffer) See example on clppng: Planar reflectons How should you render? 1) the ground plan polygon nto the stencl buffer 2) the scaled (1,1,-1) model, but mask wth stencl buffer Reflect lght pos as well Use front face cullng 3) the ground plane (sem-transparent) 4) the unscaled model 14

15 Lekton 11 Global Illumnaton Global belysnng: Why s not standard ray tracng enough? Lght transport notaton, renderng eq., BRDFs Monte Carlo Ray Tracng Path tracng Photon mappng Ray tracng L L o e f ( x,, ') L ( x, ')( ' n) d' r Global Illumnaton Lekton 12: Pertve correct texturng Taxonomy: Sort frst sort mddle sort last fragment sort last mage Bandwdth Why t s a problem How to solve t Be able to sketch the archtecture of a moder graphcs card Department of Computer Engneerng Applcaton PCI-E x16 Vertex shader Vertex shader Vertex shader Prmtve assembly Geo shader Geo shader Clppng Fragment Generaton Geo shader On NVIDIA 8-seres: Vertex-, Geometryand Fragment shaders allocated from a pool of 128 processors Fragment shader Fragment shader Fragment shader Fragment Merge Fragment Merge Fragment Merge 15