Projection Transformations. and

Transcription

1 3D Viewig Projectio Trasormatios a Viewig i Pipelie

2 Implemetatio o 3D Viewig 3-D worl cooriate output primitives Apply ormalizig i trasormatio Clip agaist caoical View Volume 2D evice cooriates Trasorm ito viewport i 2D evice cooriates or isplay Project oto Projectio plae

3 Caoical view volume or allel projectio is eie by six plaes: X = -; Y = -; Z = ; X = ; Y = ; Z = -. X or Y FP BP - -Z -

4 Caoical view volume or perspective p projectio is eie by six plaes: X = Z; Y = -Z; Z = -Z mi ; X = -Z; Y = Z; Z = -. X or Y FP - -Z - BP

5 View Speciicatios: VP, VRP, VUP, VPN, PRP, DOP, CW, VRC v VP VUP (u max, v max ) CW VRP (u mi, v mi ) VPN u

6 VUP v VP (u max, v max) ) CW VRP (u mi, v mi ) VPN u v Diagrams or a arbitrary 3D view VP VPN CW VRP u COP/PRP

7 Speciyig a Arbitrary 3D View Example Values Viewig Parameter Set Set 2 Set 3 VRP (WC) (,, 54) (6,, 54) (,, ) VPN (WC) (,, ) (,, ) (,, ) VUP (WC) (,, ) (-,, ) (,, ) PRP (VRC) (8, 6, 3) (2, 8, 3) (8, 6, 84) Wiow (-, 7, (-, 25, (-5, 5, (VRC) -, 7) -5, 2) -5, 5) Projectio Type F & B (VRC) Perspective Parallel Perspective +,

8 Steps or implemetig ormalizig trasormatio matrix or allel projectio Traslate the VRP to origi Rotate VRC such that VPN (-axis) aligs with Z-axis (also, u with X a v with Y-axis) Shear (ot ecessary or pure orthographic) such that DOP is allel to the Z-axis Traslate a scale ito allel-projectio caoical view volume (CVV) N S T SH RT( VRP)

9 Step 2 i ormalizig trasormatios: Rotate VRC such that VPN (-axis) aligs with Z-axis (also, u with X a v with Y) v VP VUP (u max, v max ) CW VRP (u mi, v mi ) VPN u

10 Expressios or Step 2 must be erive. Implemet usig the cocept o combie trasormatio (rotatio). Take R x = x cos( ) si( ) si( ) cos( ) Rows are uit vectors, whe rotate by R x, will alig with the Y a Z axis respectively. Whe uit vectors alog the priciple axes are rotate by R x, they orm the colum vectors.

11 R x = cos( ) si( ) si( ) cos( ) Z Row Vectors: [ ] [ cos() -si() ] [ si() cos() ] Colum Vectors {cosier R x (-), i this case}: [ ] T Y [ cos() -si() ] T [ si() cos() ] T

12 Y P P 3 P 2 Cosier a geeral sceario o combie rotatios a use the property erive base o the orthogoality o the R matrix. X Y Z P Beore Trasormatio P 3 Z P 2 X P P Ater 2 Trasormatio

13 Let the eective rotatio matrix be a combiatio o three rows as: where, r r r x 2x 3x r r PP T 2 y z r r 2y 2z r r 3y 3z R z= r z r 2z r 3z = PP 2 Z x x 2x 3x R = r r r = a PP X PP PP X PP T 2 3 T 2 3 P 3 Ry= r y r 2y r3y =R z X R x Z P 2 Y P Y P P P 3 P 2 P X X

14 Y P 3 P 2 Y P P 3 P X P P Z P 2 P X Z Thus the rotatio matrix o step 2 i ormalizig trasormatios, ca be ormulate as: R r r r x 2x 3x r r r y 2y 3y r r r z 2z 3z

15 Step 2 i ormalizig trasormatios: Rotate VRC such that VPN (-axis) aligs with Z-axis (also, u with X a v with Y) v VP VUP (u max, v max ) CW VRP (u mi, v mi ) VPN u

16 where, R VPN z ; VPN VUP R R z ; x VUP R z a R y R z R x The overall combie trasormatio matrix or allel projectio (WCSVV -> PPCVV), is: N S T SH RT( VRP)

17 The overall combie trasormatio matrix or allel l projectio (WCSVV -> PPCVV), is: N where, SH S T SH shx shy RT( shx shy VRP) op op op op x z y z ; y DOP VPN Sie view o -z shearig o the VV y DOP VPN -z

18 The overall combie trasormatio matrix or allel l projectio (WCSVV -> PPCVV), is: N S T SH RT( VRP) T u max u v v mi max mi FP ; 2 2 S S( ( u max 2 u mi, v max 2 v mi, FP BP )

19 Implemetig ormalizig trasormatio matrix or perspective projectio v VP CW VRP VPN u COP/PRP

20 Caoical view volume or perspective projectio is eie by six plaes: X = Z; Y = -Z; Z = -Z mi ; X = -Z; Y = Z; Z = -. X or Y FP - -Z - BP

21 Steps or implemetig e ormalizig trasormatio matrix or perspective projectio Traslate the VRP to origi i Rotate VRC such that VPN (-axis) aligs with Z-axis (also, u with X- a v with Y-axis) Traslate such that t COP (or PRP) is at the origi Shear such that ceter lie o view volume (VVCL) becomes z-axis Scale such that VV becomes the caoical view volume (CVV)

22 Sceario o the cross-sectio o the VV ater irst three trasormatios. ti X or Y CW -Z VPN VRP N per S SH T( PRP) RT( VRP) per

23 Comiso the overall combie trasormatio matrices or: PARALLEL PROJECTION: N S T SH R T( VRP) PERSPECTIVE PROJECTION: N per S per SH T( PRP) R T( VRP)

24 Implemetatio o 3D Viewig 3-D worl cooriate output primitives Apply ormalizig i trasormatio Clip agaist caoical View Volume 2D evice cooriates Trasorm ito viewport i 2D evice cooriates or isplay Project oto Projectio plae

25 Geeralize ormula o perspective projectio matrix: PP XorY (COP) L Q ( x, y, z ) O x x zp z z y y z p z z 2 Zp Zp Z Q z Q z Zp Qz Qz Mge 2 P (x p, y p, Z p ) P(X,Y,Z) Z (,, Z p ) p

26 Cooriate Systems a Matrices Perspective Parallel 3-D moelig (object) cooriates Moelig Trasormatio R.T(-VRP) RT(-VRP) R.T( View 3D Worl Orietatio Cooriates matrix ti Cot

27 View Orietatio matrix View reerece Cooriates View Mappig matrix Clip, trasorm ito 2D scree cooriates Normalize projectio Cooriates 2D evice cooriates M CVV3DVP M. S per. SH. T(-PRP) S. T. SH

28 where ater clippig, use M CVV3DVP = T(X vmi, Y vmi, Z vmi ). X X Y Y S( vmax vmi, vmax vmi, Z Z vmax vmi 2 2. T() T(,,) )

29 The 3D Viewig Pipelie Objects are moele i object (moelig) space. Trasormatios are applie to the objects to positio them i worl space. View ameters are speciie to eie the view volume o the worl, a projectio plae, a the viewport o the scree.

30 Objects are clippe to this View volume. The results are projecte oto the projectio plae (wiow) a ially mappe ito the 3D viewport. Hie objects are the remove. The objects are sca coverte a the shae i ecessary.

31 Flowchart o the 3D Viewig Pipelie Moel Object Space Scale,Rotate, Object Traslate Objects Apply Normalizig Trasormatio Clip Worl Worl Space Space Speciy View, Calculate Normal, Illumiatio, Backace Eye Space Cot

32 Eye Space Perspective Trasormatio /Projectio Image Space Remove Hie Suraces Map to Device Viewport/ Shae, Cooriates Draw to Texture scree

33 The Computer Graphics Pipelie Viewig Process

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35 The Camera Moel We speciy our iitial camera moel by ietiyig the ollowig ameters.. A scee, e, cosistig st o polygoal elemets e e each represete by their vertices; 2. A poit that t represets the camera positio: C = [C x, C y, C z ]; 3. A poit that represets the ceter-o attetio o the camera (i.e. where the camera is lookig): A = [A x, A y, A z ]; 4. A iel-o-view agle,,represetig esetig the agle subtee at the apex o the viewig pyrami.

36 The speciicatio o ear a ar bouig plaes. These plaes cosiere perpeicular to the irectio-o-view vector are at a istace o a rom the camera, respectively. C A Far Plae

37 The Viewig Pyrami 3D view o the viewig space A v Far w C u Plae The image space volume: u u, v, w

38 Sie view o the viewig space Near Plae Fa r Pla ae

39 Derivatio o the viewig trasormatio matrix, i terms o camera ameters:.u.v.u.v.w (u,v,w) (,, ) (,, ) w w w w w Thus, (u, v, w, ) (.u,.v,.w, w) u or v PP P(u, v, w) (u, v, w ) O (COP) -w

40 Express as trasormatio: ) or ;( P ) or ;( P w.w.v.u w v u

41 Trasormatio o the iite (trucate) viewig pyrami to the The image space cube (CVV), - < u, v, w <. volume: Let us irst aalyze w-axis oly. Use the trasormatio matrix: u, v, w P a b ; such that, (,, -)P (,, ) a (,, -)P (,, -) Solve or ameters a a b, usig the p, g above equatios:

42 From the costraits o the above two equatios: a b a a a. b The solutio: a ; b 2.

43 Hece the trasormatio is: P 2. Wh t b t i t ti What about u a v-axis trasormatios i the pyrami?

44 u a v-axis trasormatios i the pyrami u or v.ta(/2) ta(/2).ta(/2) /2 O (COP) (,, -) (,, -) -w

45 Trasormatios or the two poits are as ollows: v (,.ta(/2), -) (,.ta(/2), -) /2 O (COP) (,, -) (,, -) -w

46 Desire ormalize 3-D cooriates or both the 2 P poits: [,, +/-, ]. 2.. /2).ta( P 2 /2).ta( /2).ta(

47 Thus moiy P to be: 2) / ( 2) cot( / 2) / cot( P' 2 P 2. ' /2) ta( P 2. /2).ta( P

48 Its iverse has the orm: P ta ( / 2) ta( / 2) 2 2

49 P P P The Viewig Trasormatio Matrix 2) / (.cot.p P P 2 ) ( 2) / (.cot ) ( ) or ;( P cot( ) cot( 2) / 2 / ) or ;( P 2 P w w v u w v u 2. w.w.v.u w v u

50 2) / cot( or usig the regular expressio o P 2) / cot( 2) / cot( ) ( 2 - ) ( ) ( ) or ;( P cot( ) cot( 2) / 2 / ) or ;( P 2 P w w v u w v u 2. w.w.v.u w v u

51 E o Lectures o 3D Viewig Projectio Trasormatios a Viewig i Pipelie