Rendering. Ray Tracing

Transcription

1 CS475m - Compter Graphics Lectre 16 : 1 Rederig Drawig images o the compter scree. We hae see oe rederig method already. Isses: Visibility What parts of a scee are isible? Clippig Cllig (Backface ad Occlsio) Illmiatio Tracig the rays of light to model the iteractio of light with objects. Reflectio, Refractio, Shadows, etc. ay rays will ot cotribte to the image! Ierse or Backward Basic Idea For eery piel i the image Shoot a ray Fid closest itersectio with object Fid ormal at the poit of itersectio Compte illmiatio at poit of itersectio Assig piel color Primary ad Secodary Rays Viewer View Plae Trace rays from the camera (eye) throgh eery piel o the image to the world.

2 Shadow Rays Ray Castig Visibility Ray Object Itersectios Illmiatio Piel Color determiatio (shadig) Viewer Viewer View Plae View Plae Ray Represetatio Ray Sphere Itersectio Ray Sphere Itersectio =[ c y c z c ] T Ceter =[ d y d z d ] T R 0 =[ o y o z o ] T Ray Origi Ray Directio Radis [ s y s z s] T Srface Poit We get a adratic i t At 2 BtC=0 d 2 y d 2 z d 2 =1 Parametric form: Rt=R 0 t, t0 Implicit form of the sphere: s c 2 y s 2 z s 2 2 = To sole for the eatio we sbstitte: where A= 2 d y 2 d z 2 d =1 B=2 d o c y d y o z d z o C= o c 2 y o 2 z o t d c 2 y 0 t y d 2 z 0 t z d 2 = 2

3 Ray Sphere Itersectio Ray Sphere Itersectio Ray Sphere Itersectio Solig for t B t 0 = B2 4AC B 2A t 1 = B2 4AC 2A Smallest positie ale amog these two itersectios is the closest itersectio poit. i, y i, z i = 0 t d, y 0 t y d, z 0 t z d N i = N i The ormal at the poit of itersectio is gie by: i c, y i y c The steps are: Calclate A, B, C Compte the discrimiat Calclate miimm t. Compte itersectio poit. Compte the ormal., z z i c L t CO Sr d t ' Geometrically: L= t CO =L T If t CO 0 the o itersectio. d 2 =L L T 2 t CO If d the o itersectio. t '= 2 d 2 The the two itersectios are gie by t 0 =t CO t ' ad t 1 =t CO + t ' Ray Plae Itersectio Ray Plae Itersectio Ray Plae Itersectio Ray Rt =R 0 t, t0 Sbstitig A o t d By o ty d C z o tz d D=0 Sbstitig A o t d By o ty d C z o tz d D=0 Plae Plae Normal P: AByCzD=0 A 2 B 2 C 2 =1 P = A, B,C D: Distace from the origi Solig: If: t= A By Cz D o o o = P. R D o A d By d Cz d P. V d = A d By d Cz d =P. Now, if V d =0 the ray is parallel to the plae (i.e., o itersectio). Now, if V d 0 the ormal is poitig away from the ray. Ca be sed for backface cllig. Solig: If: V o = A o By o Cz o D= P. D t= V o the V d If t0 the plae is behid ray's origi, else compte itersectio. t= A By Cz D o o o = P. R D o A d By d Cz d P. i, y i, z i = 0 t d, y 0 t y d, z 0 t z d N i =P

4 Ray Polygo Itersectio Do a Ray Plae itersectio ad the check for cotaimet i the polygo. Ray Triagle Itersectio Do a Ray Plae itersectio ad the check for cotaimet i the triagle. Ray Qadric Itersectio Cyliders, Coes, Sphere, Ellipsoids, Paraboloids Shoot a ray i ay directio from the itersectio poit i the plae of the polygo. If mber of itersectios with the polygo bodary are odd the the poit is cotaied i the polygo V 2 V 1 The poit of itersectio ca be writte i Barycetric Coordiates as: A A P 2 3 P= V 1 V 2 wv 3 A V 1 3 where = A 1 A, = A 2 A, w= A 3 A ad A= A 1 A 2 A 3 The poit lies iside the triagle if w=1 ad 0, 0, w 0 Implicit form of a geeral adric is gie by: F, y, z = A 2 2By2Cz2DEy 2 2Fyz2GyHz 2 2IzJ=0 Ray: Rt=R 0 t, t0 Sbstitte ad sole the adratic i t Normal at the poit of itersectio: N i = F i, F y i, F z i N i =2A i By i CZ i D N iy =2 B i Ey i FZ i G N iz =2C i Fy i HZ i I Ray Bo Itersectio User Cyrs Beck/Liag Barsky i 3D Settig p Viewig y o WCS z p E R VCS I WCS Positio of VRP, R Normal to View Plae, Up Vector, I VCS p Etet of widow Positio of Eye, E Settig p Viewig ' p p If ' p = p p. the we defie = ' p ' p ad =

5 Settig p z Viewig O y P If a poit P has coordiates, y, z i WCS a, b, c i VCS the a y b [ z ]=.[ c]r E e, e, e Rr,r y, r z where, =[ y y y z z ] z Settig p Viewig E e, e, e P * *, * Defie VRP, Rr,r y, r z Compte from,, Defie eye i VCS E e, e, e =0, 0, e I WCS, the eye becomes E wcs =. ER P p, p, p Settig p Widow E 0, 0, e W t W b W l W r Defie widow size W l,w r,w t,w b * i =W l i for piel i, j * i =W t j where =W r W t / AXCOLS =W t W b / AXROWS Ths coordiates of piel i, j are P * ij = * i, * j, 0 AXROWS ad AXCOLS ca be sed to cotrol the resoltio of the image. Settig p E wcs =. E R=. [ 0 0 e ] T R Basic Idea Ray W t W b Directio of the ray from the eye throgh piel at (i,j) i WCS is gie by dir ij =.P * ij R E wcs =.P * ij E =. [ i * j * e ] T Ray is throgh piel i, j gie by R ij t =E wcs dir ij t= t For eery piel i the image Shoot a ray Fid closest itersectio with object Fid ormal at the poit of itersectio Compte illmiatio at poit of itersectio Assig piel color Ray: sc t Objects: Sphere, Coe, Cylider, Bo We assme the objects to be ormalized so that the ray object itersectio is easier: for e.g., Sphere: 2 y 2 z 2 =1 E 0, 0, e W l W r

6 ' Ray: sc t ' s 'c ' t ' Objects: Sphere, Coe, Cylider, Bo The we replicate the ormalized objects ad trasform them to create ariety i the scee. Ca ray object itersectios be doe o the trasformed objects? Object Space 1 A ormalized sphere is trasformed der a affie trasformatio, i.e., '=. = 1. ' 1 Object Space sc t sct= 1. s' 1.c ' t ' [ ]=[ s a d g l ][ ]=[ s' as ' ds ' ygs' zl s y b e h m s' y bs ' es' y hs' z ] m s z c f i s' z cs' fs ' y is ' z A side ote o trasformatios Projectie Affie ' s' c ' t ' ' s 'c ' t ' Similarity/Coformal Rigid/Eclidea Liear Object Space [c c y c z 0 sc t 1 ]=[ ]=[a d g l ' b e h m c ' y c f i ][c sct= 1. s' 1. c' t ' ac ' dc ' ygc' zl bc ' ec ' y hc ' z ] m c ' z cc ' fc ' y ic ' z 0 0 Object Space sc t 1 If icldes a scalig the c is ot a it ector after the trasformatio. If c is reormalized the scale the tale accordigly. Traslatio Idetity Rotatio Perspectie Isotropic Scalig Scalig Reflectio Shear Orthogoal

7 N ' ' s' c ' t ' Object Space N sc t 1 We also eed the ormal at the poit of itersectio. How does the ormal trasform whe the object dergoes a affie trasformatio?