Modeling Transformations

Transcription

1 Modeling Transformations Michael Kazhdan ( /657) HB Ch. 5 FvDFH Ch. 5

2 Overview Ra-Tracing so far Modeling transformations

3 Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int heigh, int depth, float cutoff) { Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) { Ra ra = ConstructRaThroughPiel(camera, i, j); image[i][j] = GetColor( scene, ra, 1, depth, Color( cutoff, cutoff, cutoff ) ); } return image; }

4 Ra Tracing For each sample Construct ra from ee position through view plane Compute color contribution of the ra Ras through view plane Ee position Samples on view plane

5 Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int height, int depth, float cutoff) { Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) { Ra ra = ConstructRaThroughPiel(camera, i, j); image[i][j] = GetColor( scene, ra, 1, depth, Color( cutoff, cutoff, cutoff ) ); } return image; }

6 Constructing Ra Through a Piel 2D Eample: The ra passing through the i-th piel is defined b: Ra t = p 0 + t v p 0 : camera position v: direction to the i-th piel: v = p i p 0 p i p 0 p 0 θ p[i]: i-th piel location: v i p i = p d tan θ up height p 1 and p 2 are the endpoints of the view plane: p 1 = p 0 + d towards d tan θ up p 2 = p 0 + d towards + d tan θ up up towards d p[i] p 2 p 1 2 d tan(θ)

7 Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int height, int depth, float cutoff) { Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) { Ra ra = ConstructRaThroughPiel(camera, i, j); image[i][j] = GetColor( scene, ra, 1, depth, Color( cutoff, cutoff, cutoff ) ); Color GetColor( } Scene scene, Ra ra, float ir, int d, Color co ) return image; { } HitInformation hit; Color c(0,0,0); if( d 0 (co[0]>1 && co[1]>1 && co[2]>1) ) return c; } if( Intersect( scene, ra, hit ) 0 ) { c = GetAmbientAndEmissive( hit ); for( i=0 ; i<lightnum ; i++) c += light[i].getdiffuseandspecular( ra, hit ); Ra reflectedra = ReflectedRa( ra, hit.p, hit.n ); c += GetColor( scene, reflectedra, ir, d-1, co/hit.spec ) * hit.spec; Ra refractedra = RefractedRa( ra, hit.p, hit.n, hit.ir); c += GetColor( scene, refractedra, hit.ir, d-1, co/hit.tran ) * hit.tran; } return c;

8 Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int height, int depth, float cutoff) { Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) { Ra ra = ConstructRaThroughPiel(camera, i, j); image[i][j] = GetColor( scene, ra, 1, depth, Color( cutoff, cutoff, cutoff ) ); Color GetColor( } Scene scene, Ra ra, float ir, int d, Color co ) return image; { } HitInformation hit; Color c(0,0,0); if( d 0 (co[0]>1 && co[1]>1 && co[2]>1) ) return c; } if( Intersect( scene, ra, hit ) 0 ) { c = GetAmbientAndEmissive( hit ); for( i=0 ; i<lightnum ; i++) c += light[i].getdiffuseandspecular( ra, hit ); Ra reflectedra = ReflectedRa( ra, hit.p, hit.n ); c += GetColor( scene, reflectedra, ir, d-1, co/hit.spec ) * hit.spec; Ra refractedra = RefractedRa( ra, hit.p, hit.n, hit.ir); c += GetColor( scene, refractedra, hit.ir, d-1, co/hit.tran ) * hit.tran; } return c;

9 Ra-Scene Intersection Intersections with geometric primitives Sphere Triangle Groups of primitives (scene) Acceleration techniques Bounding volume hierarchies Spatial partitions» Uniform grids» Octrees» BSP trees

10 Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int height, int depth, float cutoff) { Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) { Ra ra = ConstructRaThroughPiel(camera, i, j); image[i][j] = GetColor( scene, ra, 1, depth, Color( cutoff, cutoff, cutoff ) ); Color GetColor( } Scene scene, Ra ra, float ir, int d, Color co ) return image; { } HitInformation hit; Color c(0,0,0); if( d 0 (co[0]>1 && co[1]>1 && co[2]>1) ) return c; } if( Intersect( scene, ra, hit ) 0 ) { c = GetAmbientAndEmissive( hit ); for( i=0 ; i<lightnum ; i++) c += light[i].getdiffuseandspecular( ra, hit ); Ra reflectedra = ReflectedRa( ra, hit.p, hit.n ); c += GetColor( scene, reflectedra, ir, d-1, co/hit.spec ) * hit.spec; Ra refractedra = RefractedRa( ra, hit.p, hit.n, hit.ir); c += GetColor( scene, refractedra, hit.ir, d-1, co/hit.tran ) * hit.tran; } return c;

11 Surface Illumination Calculation Viewer L 1 L 2 N V I = I E + L K A I L A + K D N, L + K S V, R n I L S L

12 Ra Tracing (Recursive) Image RaTrace(Camera camera, Scene scene, int width, int height, int depth, float cutoff) { Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) { Ra ra = ConstructRaThroughPiel(camera, i, j); image[i][j] = GetColor( scene, ra, 1, depth, Color( cutoff, cutoff, cutoff ) ); Color GetColor( } Scene scene, Ra ra, float ir, int d, Color co ) return image; { } HitInformation hit; Color c(0,0,0); if( d 0 (co[0]>1 && co[1]>1 && co[2]>1) ) return c; } if( Intersect( scene, ra, hit ) 0 ) { c = GetAmbientAndEmissive( hit ); for( i=0 ; i<lightnum ; i++) c += light[i].getdiffuseandspecular( ra, hit ); Ra reflectedra = ReflectedRa( ra, hit.p, hit.n ); c += GetColor( scene, reflectedra, ir, d-1, co/hit.spec ) * hit.spec; Ra refractedra = RefractedRa( ra, hit.p, hit.n, hit.ir); c += GetColor( scene, refractedra, hit.ir, d-1, co/hit.tran ) * hit.tran; } return c;

13 Ra Tracing (Recursive) Consider the contribution of: Reflected Ras Refracted Ras L 1 L 0 I = I E + L K A I L A + K D N, L + K S V, R n I L S L + K s I R + K T I T

14 Overview Ratracing so far Modeling transformations

15 Modeling Transformations Specif transformations for objects Allows definitions of objects in own coordinate sstems Allows use of object definition multiple times in a scene H&B Figure 109

16 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D

17 Simple 2D Transformations Translation p = p + t p p = p p + t t p p

18 Simple 2D Transformations Translation p = p + t p p = p p + t t

19 Simple 2D Transformations Scale p = S p p p = s 0 0 s p p

20 Simple 2D Transformation Rotation p = R p p p = cos θ sin θ sin θ cos θ p p

21 2D Modeling Transformations Object Coordinates Scale Translate Scale Rotate Translate World Coordinates

22 2D Modeling Transformations Object Coordinates Let s look at this in detail World Coordinates

23 2D Modeling Transformations Object Coordinates

24 2D Modeling Transformations Object Coordinates Scale.3,.3 Rotate -90 Translate 5, 3

25 2D Modeling Transformations Object Coordinates Scale.3,.3 Rotate -90 Translate 5, 3

26 2D Modeling Transformations Object Coordinates Scale.3,.3 Rotate -90 Translate 3, 5 World Coordinates

27 Basic 2D Transformations Translation: = + t = + t Scale: = s = s Rotation: = cos θ sin θ = sin θ + cos θ Transformations can be combined (with simple algebra)

28 Basic 2D Transformations Translation: = + t = + t Scale: = s = s Rotation: = cos θ sin θ = sin θ + cos θ

29 Basic 2D Transformations Translation: = + t = + t Scale: = s = s (, ) (, ) Rotation: = cos θ sin θ = sin θ + cos θ = s = s

30 Basic 2D Transformations Translation: = + t = + t Scale: = s = s Rotation: = cos θ sin θ = sin θ + cos θ (, ) = s = s = s cos θ s sin θ = s sin θ + s cos θ

31 Basic 2D Transformations Translation: = + t = + t Scale: = s = s (, ) Rotation: = cos θ sin θ = sin θ + cos θ = s = s = s cos θ s sin θ = s sin θ + s cos θ = s cos θ s sin θ + t = s sin θ + s cos θ + t

32 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D

33 Matri Representation Represent 2D transformation b a matri a c b d Multipl matri b column vector appl transformation to point = a b c d = a + b = c + d

34 Matri Representation Transformations combined b multiplication = a b c d e g f h i k j k Matrices are a convenient and efficient wa to represent a sequence of transformations!

35 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Scale around (0,0)? = s = s = s 0 0 s 2D Rotate around (0,0)? = cos θ sin θ = sin θ + cos θ = cos θ sin θ sin θ cos θ 2D Mirror over Y ais? = = =

36 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Scale around (0,0)? = s = s Like scale with negative scale values = s 0 0 s 2D Mirror over Y ais? = = =

37 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation?

38 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation? = + t = + t ' ' NO! a c b d Onl linear 2D transformations can be represented with a 2 2 matri

39 Linear Transformations Linear transformations are combinations of Scale, and Rotation Properties of linear transformations: Satisfies: T s 1 p 1 + s 2 p 2 = s 1 T p 1 + s 2 T(p 2 ) Origin maps to origin Lines map to lines Parallel lines remain parallel Closed under composition = a b c d

40 Linear Transformations Linear transformations are combinations of Scale, and Rotation Properties of linear transformations: Satisfies: T s 1 p 1 + s 2 p 2 = s 1 T p 1 + s 2 T(p 2 ) Origin maps to origin = Lines map to lines Parallel lines remain parallel Closed under composition a b c d Translations do not map the origin to the origin

41 2D Translation 2D translation represented b a 33 matri Point represented with homogeneous coordinates = + t w = + t w w = w w = 1 0 t 0 1 t w

42 2D Translation 2D translation represented b a 33 matri Point represented with homogeneous coordinates = + t = + t w = 1 1 = 1 0 t 0 1 t Fiing w = 1, we can get translation.

43 Homogeneous Coordinates Add a 3rd coordinate to ever 2D point (,, w) represents a point at location (,, 0) represents a point at infinit (0, 0, 0) is not allowed 2 1, w w (2,1,1) or (4,2,2) or (6,3,3) 1 2 Convenient coordinate sstem to represent man useful transformations

44 Basic 2D Transformations Basic 2D transformations as 3 3 matrices t = 0 1 t Translate 1 1 = s s Scale 1 1 = cos θ sin θ 0 sin θ cos θ Rotate 1

45 Affine Transformations Affine transformations are combinations of Linear transformations, and Translations w = a b c d e f Properties of affine transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines remain parallel Closed under composition w

46 Projective Transformations Projective transformations Affine transformations, and Projective warps w = a b c d e f g h i Properties of projective transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines do not necessaril remain parallel Closed under composition w

47 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D

48 Matri Composition Transformations can be combined b matri multiplication w = 1 0 t 0 1 t cos θ sin θ 0 sin θ cos θ s s p = T t, t R θ S s, s p w

49 Matri Composition Matrices are a convenient and efficient wa to represent a sequence of transformations General purpose representation Hardware matri multipl Efficienc with pre-multiplication» Matri multiplication is associative p = T R S p p = (T R S)(p)

50 Matri Composition Be aware: order of transformations matters» Matri multiplication is not commutative p = T R S p Global Local

51 Matri Composition Rotate b θ around arbitrar point (a, b) M = T a, b R θ T( a, b) The trick: First, translate (a, b) to the origin. Net, do the rotation about origin. Finall, translate back. (a, b) Scale b (s, s ) around arbitrar point (a, b) M = T a, b S s, s T( a, b) (Use the same trick.) (a, b)

52 Overview 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D

53 3D Transformations Same idea as 2D transformations Homogeneous coordinates: (,, z, w) 4 4 transformation matrices z w = a b c d e f g h i j k l m n o p z w

54 Basic 3D Transformations z w = Identit z w z w = s s s z Scale z w z w = t t t z z w Translation

55 Basic 3D Transformations Pitch-Roll-Yaw Convention: An rotation can be epressed as the combination of a rotation about the -, the -, and the z-ais. Rotate around z ais: z w = cos θ sin θ 0 0 sin θ cos θ z w Rotate around ais: z w = cos θ 0 sin θ sin θ 0 cos θ z w Rotate around ais: z w = cos θ sin θ 0 0 sin θ cos θ z w

56 Basic 3D Transformations Pitch-Roll-Yaw Convention: An rotation can be epressed as the combination of a rotation about the -, the -, and the z-ais. Rotate around z ais: z w = cos θ sin θ 0 0 sin θ cos θ z w How do ou rotate around an Rotate around ais: = arbitrar ais U b angle ψ? Rotate around ais: z w z w = cos θ 0 sin θ sin θ 0 cos θ cos θ sin θ 0 0 sin θ cos θ z w z w

57 Rotation B ψ Around Arbitrar Ais U Align U (wlog) with the z-ais: T (a,b,c) : Translate U b (a, b, c) to pass through origin R φ R θ : Rotate around the - and -aes b θ and φ degrees to get U aligned with the z-ais. R z (ψ): Perform rotation b ψ around the z-ais. Do inverse of original transformation for alignment. U U z T (a,b,c) z R φ R θ z U

58 Rotation B ψ Around Arbitrar Ais U Align U (wlog) with the z-ais: T (a,b,c) : Translate U b (a, b, c) to pass through origin R φ R θ : Rotate around the - and -aes b θ and φ degrees to get U aligned with the z-ais. R z (ψ): Perform rotation b ψ around the z-ais. Do inverse of original transformation for alignment. p = R φ R θ T a,b,c 1 Rz ψ R φ R θ T a,b,c p Aligning Transformation