University of British Columbia View Volumes Canonical View Volumes Why Canonical View Volumes? CPSC 314 Computer Graphics Jan-Apr 2016 • specifies field-of-view, used for clipping • standardized viewing volume representation • permits standardization • restricts domain of z stored for visibility test • clipping Tamara Munzner perspective orthographic • easier to determine if an arbitrary point is perspective view volume orthographic view volume orthogonal enclosed in volume with canonical view y=top parallel volume vs. clipping to six arbitrary planes y=top x=left x=left • rendering y y x or y x or y = +/- z x or y back • projection and rasterization algorithms can be Viewing 3 z plane z x=right back 1 reused VCS front plane front VCS y=bottom z=-near z=-far x -z x z=-far plane -z plane -1 x=right y=bottom z=-near http://www.ugrad.cs.ubc.ca/~cs314/Vjan2016 -1 2 3 4

Normalized Device Coordinates Normalized Device Coordinates Understanding Z Understanding Z

near, far always positive in GL calls • convention left/right x =+/- 1, top/bottom y =+/- 1, near/far z =+/- 1 • z axis flip changes coord system handedness THREE.OrthographicCamera(left,right,bot,top,near,far); • viewing frustum mapped to specific mat4.frustum(left,right,bot,top,near,far, projectionMatrix); • RHS before projection (eye/view coords) parallelepiped Camera coordinates NDC • LHS after projection (clip, norm device coords) • Normalized Device Coordinates (NDC) x x perspective view volume orthographic view volume • same as clipping coords x=1 VCS NDCS y=top • only objects inside the parallelepiped get right y=top y (1,1,1) y=top x=left x=left x=left rendered Frustum y -z z z y left y • which parallelepiped? z (-1,-1,-1) z x= -1 x=right x x=right VCS • depends on rendering system z= -1 z=1 x z=-far VCS y=bottom z=-near z=-far x x z=-far y=bottom y=bottom z=-n z=-f z=-near x=right z=-near 5 6 7 8

Understanding Z Orthographic Derivation Orthographic Derivation Orthographic Derivation

• why near and far plane? • scale, translate, reflect for new coord sys • scale, translate, reflect for new coord sys • scale, translate, reflect for new coord sys • near plane: y = top → y'= 1 y = top → y'= 1 1 = a ⋅top + b y'= a ⋅ y + b y'= a ⋅ y + b • avoid singularity (division by zero, or very y = bot → y' = −1 y = bot → y' = −1 −1 = a ⋅bot + b small numbers) VCS NDCS VCS NDCS 2 • far plane: 1 = top + b b = 1− a⋅top,b = −1− a ⋅bot top bot y=top y y=top y − • store depth in fixed-point representation x=left (1,1,1) x=left (1,1,1) 1 a top 1 a bot y y − ⋅ = − − ⋅ 2⋅top (integer), thus have to have fixed range of z z b = 1− z (-1,-1,-1) z (-1,-1,-1) 1− (−1) = −a⋅bot − (−a⋅top) top − bot values (0…1) x=right x x=right x 2 = a(−bot + top) (top − bot ) − 2⋅top x x b = • avoid/reduce numerical precision artifacts for z=-far z=-far 2 top − bot distant objects y=bottom z=-near y=bottom z=-near a = − top − bot top − bot b = top bot 9 10 11 − 12

Orthographic Derivation Orthographic Derivation Orthographic Derivation Orthographic Derivation

• scale, translate, reflect for new coord sys • scale, translate, reflect for new coord sys • scale, translate, reflect for new coord sys • scale, translate, reflect for new coord sys y = top → y'= 1 y'= a ⋅ y + b  2 right + left   2 right + left   2 right + left  y = bot → y' = −1 0 0 − 0 0 − 0 0 − right − left right − left  right − left right − left  right − left right − left        VCS 2 a =  2 top + bot   2 top + bot   2 top + bot   0 0 −   0 0 −   0 0 −  y=top top − bot top − bot top − bot top − bot top − bot top − bot top − bot x=left P' =  P P' =  P P' =  P y top + bot  − 2 far + near   − 2 far + near   − 2 far + near   0 0 −   0 0 −   0 0 −  z b = − far − near far − near far − near far − near far − near far − near x=right top − bot             x z=-far  0 0 0 1   0 0 0 1   0 0 0 1  y=bottom z=-near same idea for right/left, far/near

13 14 15 16 Orthographic Derivation Projective Rendering Pipeline Projection Warp Perspective Warp

• scale, translate, reflect for new coord sys • warp perspective view volume to orthogonal • perspective viewing frustum transformed to object world viewing view volume cube O2W W2V V2C OCS WCS VCS  2 right + left  0 0 − modeling viewing projection • render all scenes with orthographic projection! • orthographic rendering of cube produces same right − left right − left  transformation transformation transformation   clipping • aka perspective warp image as perspective rendering of original  2 top + bot  C2N frustum 0 0 − OCS - object/model coordinate system CCS  top bot top bot  perspective P' =  − − P WCS - world coordinate system divide normalized x x  − 2 far + near   0 0 −  VCS - viewing/camera/eye coordinate N2D device far − near far − near system NDCS z=α z=d z=0 z=d   viewport   CCS - clipping coordinate system  0 0 0 1  transformation NDCS - normalized device coordinate device system Z Z DCS DCS - device/display/screen coordinate 17 system 18 19 20

Predistortion Projective Rendering Pipeline Separate Warp From Homogenization Perspective Divide Example

viewing clipping normalized • specific example V2C CCS C2N device object world viewing VCS O2W W2V V2C NDCS • assume image plane at z = -1 OCS WCS VCS projection perspective modeling viewing projection transformation division • a point [x,y,z,1]T projects to [-x/z,-y/z,-z/z,1]T ≡ transformation transformation transformation alter w / w clipping [x,y,z,-z]T C2N OCS - object/model coordinate system CCS • warp requires only standard matrix multiply perspective x  x  WCS - world coordinate system  y divide normalized • distort such that orthographic projection of distorted  y       z VCS - viewing/camera/eye coordinate device objects is desired persp projection  z    N2D   1 system NDCS − z • w is changed viewport CCS - clipping coordinate system • clip after warp, before divide -z transformation NDCS - normalized device coordinate device • division by w: homogenization system DCS DCS - device/display/screen coordinate 21 system 22 23 24

Perspective Divide Example Perspective Normalization Perspective To NDCS Derivation Perspective Derivation • matrix formulation  x' 1 0 0 0 x x 1 0 0 0 x x x / z 1 0 0 0   x   x                 −  x   simple example earlier: y' 0 1 0 0 y       y  z / d   =              0 1 0 0 x p     y  0 1 0 0 y y − y / z   y     y  z'  0 0 1 0 z T   =   ⋅   =   ≡   d −α ⋅d    (z − α )⋅d  y          0 0  =    p  =  w' 0 0 1/d 0 1 z 0 0 1 0 z  z   − 1   d − α d − α  z  d −α   z / d       z  2 VCS y=top              1      p   d  α  NDCS 1 0 0 −1 0 1 − z 1 0 0 0 1 z  1−                   d z complete: shear, scale, projection-normalization  d   d   −α   x=left y (1,1,1) € • after homogenizing, once again w=1 y z • warp and homogenization both preserve  x ′  E 0 A 0 x (-1,-1,-1) x      z y ′ 0 F B 0 y projection z=-near perspective relative depth (z coordinate) y=bottom z=-far   =    transformation division x  z ′   0 0 C D z x=right      alter w / w  w ′  0 0 −1 0 1

25 27 28

€

Perspective Derivation Perspective Derivation Perspective Derivation Perspective Derivation

x = left → x′ / w′= −1  x' 1 0 0 0 x  x' E 0 A 0 x x'= Ex + Az • similarly for other 5 planes  x' 1 0 0 0 x           x = right → x′/ w′ = 1 earlier: y' 0 1 0 0 y earlier:      y' 0 F B 0 y y'= Fy + Bz   =    y' 0 1 0 0 y   =    • 6 planes, 6 unknowns  z'  0 0 1 0 z   =     z'   0 0 C D z y = top → y′ / w′= 1 z' 0 0 1 0 z z'= Cz + D                 w' 0 0 1/d 0 1      w' 0 0 −1 0 1 y = bottom → y′ / w′= −1  2n r + l   w' 0 0 1/d 0 1      w'= −z z near z / w 1 0 0 = − → ′ ′ = −  r − l r − l  complete: shear, scale, projection-normalization complete: shear, scale, projection-normalization z = − far → z′ / w′ = 1  2n t + b  € € € €y' Fy + Bz Fy + Bz Fy + Bz  0 0  y'= Fy + Bz, = , 1= , 1= , t − b t − b  x ′  E 0 A 0 x  x ′  E 0 A 0 x w' w' w' −z             −( f + n) −2 fn y ′ 0 F B 0 y y ′ 0 F B 0 y y z y top     =      =    1 = F + B , 1 = F − B, 1= F − B, 0 0  z ′   0 0 C D z  z ′   0 0 C D z −z −z −z −(−near)  f − n f − n            w ′  0 0 −1 0 1  w ′  0 0 −1 0 1 top 1 = F − B  0 0 −1 0  near 29 30 31 32 € € € € € Projective Rendering Pipeline NDC to Device Transformation Origin Location N2D Transformation

• map from NDC to pixel coordinates on display • yet more (possibly confusing) conventions • general formulation object world viewing • NDC range is x = -1...1, y = -1...1, z = -1...1 • GL origin: lower left • reflect in y for upper vs. lower left origin O2W W2V V2C OCS WCS VCS • typical display range: x = 0...500, y = 0...300 • most window systems origin: upper left • scale by width, height, depth modeling viewing projection transformation • maximum is size of actual screen transformation transformation clipping • then must reflect in y • translate by width/2, height/2, depth/2 C2N • z range max and default is (0, 1), use later for visibility • FCG includes additional translation for pixel centers at OCS - object/model coordinate system CCS • when interpreting mouse position, have to flip your y perspective gl.viewport(0,0,w,h); (.5, .5) instead of (0,0) gl.depthRange(0,1); // depth = 1 by default coordinates WCS - world coordinate system divide normalized

device 0 x 500 0 x 500 0 x 500 VCS - viewing/camera/eye coordinate N2D y y y system NDCS 0 y 0 y 0 y height viewport CCS - clipping coordinate system -1 -1 1 transformation NDCS - normalized device coordinate device x viewport x viewport x viewport system NDC NDC NDC DCS 1 1 -1 DCS - device/display/screen coordinate width system 33 -1 1 300 34 -1 1 300 35 -1 1 300 36

N2D Transformation Device vs. Screen Coordinates Projective Rendering Pipeline Coordinate Systems

 width 1  w idth   width(xN + 1) −1  1 0 0 −  0 0 0   viewing  xD  2 2 2 1 0 0 0 x N  2      • viewport/window location wrt actual display not available glVertex3f(x,y,z) (4-space, W=1)   height 1 height     height(− yN +1) −1 yD 0 1 0 − 0 0 0 0 −1 0 0 yN   =       =   within GL z  2 2 2  0 0 1 0 z  2  object world viewing  D  depth depth    N  depth(z +1) clipping   0 0 1  0 0 0     N  • usually don’t care O2W W2V V2C alter w  1   2  2  0 0 0 1  1   2  OCS WCS VCS glFrustum(...) projection (4-space parallelepiped, projection 0 0 0 1  0 0 0 1  1  • use relative information when handling mouse events, not modeling viewing matrix with COP moved transformation Display z range is 0 to 1. absolute coordinates transformation transformation clipping backwards to infinity gl.depthRange(n,f) can • could get actual display height/width, window offsets from OS glTranslatef(x,y,z) gluLookAt(...) C2N / w CCS normalized device € glRotatef(a,x,y,z) reminder: constrain further, but depth = 1 • loose use of terms: device, display, window, screen... divide by w (3-space .... perspective NDC z range is -1 to 1 is both max and default 0 x 1024 parallelepiped) OCS - object coordinate system division normalized 0 x 500 0 0 y glutInitWindowSize(w,h) device y x 500 y offset N2D scale & device 0 y WCS - world coordinate system glViewport(x,y,a,b) NDCS height 0 y translate (3-space 1 x offset display VCS - viewing coordinate system viewport parallelipiped) viewport viewport height transformation viewport CCS - clipping coordinate system x 300 display device NDC framebuffer -1 NDCS - normalized device coordinate system DCS width 768 37 38 39 40 -1 1 300 display width DCS - device coordinate system

Perspective Example Perspective Example Perspective Example

view volume tracks in VCS: view volume  1  1  1  left = -1, right = 1 • left = -1, right = 1      left x=-1, y=-1 bot = -1, top = 1 −1 1 −1 right x=1, y=-1 • bot = -1, top = 1   =    near = 1, far = 4 • near = 1, far = 4 − 5zVCS /3− 8/ 3  −5/ 3 −8/3 zVCS   2n r + l       0 0 1 0 0 0   −zVCS   −1  1  x=-1 x=1  r − l r − l  1 ymax-1  2n t + b     0 0  0 1 0 0 z=-4 t − b t − b   real   xND CS = −1/ zVCS midpoint −( f + n) −2 fn z=-1 -1  0 0  0 0 −5/3 −8 /3 / w -1 1 0 xmax-1 x -1 0 f n f n € yND CS = 1/zVCS NDCS DCS  − −      0 0 1 0 (z not shown) (z not shown)  0 0 −1 0   −  5 8 z z = + VCS ND CS 3 3z top view VCS 41 42 43 € € €