General Purpose Computation (CAD/CAM/CAE) on the GPU (a.k.a. Topics in Manufacturing)

Transcription

1 ME 29-R: General Purpose Computation (CAD/CAM/CAE) on the GPU (a.k.a. Topics in Manufacturing) Sara McMains Spring 29 lecture 2

2 Toda s GPU eample: moldabilit feedback Two-part mold [The Complete Sculptor Inc.] Multipart mold [Priadarshi and Gupta, 24] 2 2

3 Mold Schematic Mold Cavit Mold Halves (credit: Rahul Khardekar) 3 3

4 2-Moldable Direction Parting Direction (credit: Rahul Khardekar) 4 4

5 Not 2-Moldable Parting Direction Undercut (credit: Rahul Khardekar) 5 5

6 Not 2-Moldable Parting Direction Undercut (credit: Rahul Khardekar) 6 6

7 Forming Undercuts Parting Direction Side core (credit: Rahul Khardekar) 7 7

8 GPU undercut detection Seconds CPU (Solidworks) Vs GPU For Undercut Detection and Highlighting CPU (Solidworks) GPU.5 Number of facets ,798 Solidworks triangles, 2 fps Our Algorithm (credit: Rahul Khardekar) 8 8

9 Toda s lecture Transformations and projective geometr 2D scales, shears, rotations, translations; homogeneous coordinates 9

10 The Graphics Pipeline Graphics State Application Vertices (3D) Transform & Light Xformed, Lit Vertices (2D) Assemble Primitives Screenspace triangles (2D) Fragments (pre-piels) Final Piels (Color, Depth) Rasterize Shade Video Memor (Tetures) CPU GPU Render-to-teture A simplified graphics pipeline (credit: Ian Buck, Trond Hagen)

11 GPU Pipeline: Transform Verte processor (multiple in parallel) Transform vertices from world space to image space Compute per-verte lighting (credit: Ian Buck, Trond Hagen)

12 Transformations and Matrices Transformations are functions Matrices are function representations Matrices represent linear transf s { 22 Matrices} { 2 D Linear Transf's} ) 2

13 What is a 2D Linear Transf? Recall from Linear Algebra: For a linear transformation T: r r r r Def : T ( a + ) = at ( ) + T ( ), r r for scalar a and vectors and. 3

14 Eample: Scale in Scale in, b 2, sa: ( 2 ( + ), + ) = ( 2 + 2, + ) = ( 2, ) + ( 2, ) 4

15 Eample: Scale in b 2 What is the graphical view?

16 Scale in b 2 (, ) (2, ) (2 (, ), ) 6

17 ( 2 ) + + 2, (2, ) ( 2 + ) + 2, (2, ) 7

18 ( ) 2( + + ), (, ) ( + ) + ), ( 2( + ) + ), (, ) 8

19 Summar on Scale Scale then add, is same as Add then scale 9

20 Matri Representation Scale in b 2: 2 2 = 2

21 Matri Representation Scale in b 2: = 2 2 2

22 Matri Representation Overall Scale b 2: 2 2 = =

23 Matri Form of Same = Add and, then scale = + Scale ( ) 4243 and, then add 23

24 What do the off diagonal elements do?

25 Off Diagonal Elements a + a = = b b + 25

26 ), ( (,) (,) (,) (,) Eample Eample T + = = S

27 Eample T (, ) = (,) (,).4 + S (,) (,) 27

28 Eample (,.4) T (, ) = (,) T(S).4 + (,.4) (,) 28

29 Outline Toda s lecture Transformations and projective geometr 2D scales, shears, rotations, translations; homogeneous coordinates 29

30 Eample 2 (,) (,) T (, ).6 = +.6 = S (,) (,) 3

31 Eample 2 T (, ) +.6 = (,) (,) S (,) (,) 3

32 Eample 2 T (, ) +.6 = (,) (.6, ) (.6, ) T(S) (,) (, ) 32

33 33 Summar Summar Shear in Shear in : : + = = + = = a a Sh b b Sh

34 Shear a = tanf rotates -ais CW b φ Sh = (,) a T (, ) f = + a +.6 = (.6, ) (.6, ) T(S) (, ) 34

35 35 ab) ( ( ab) Not commutative Not commutative + = + = b a a b b a b a

36 Outline Toda s lecture Transformations and projective geometr 2D scales, shears, rotations, translations; homogeneous coordinates 36

37 Rotate b q (,) q q (,) 37

38 Rotate b q : ais ( cos q,sinq ) sin q q { (,) cosq 38

39 Rotate b q : ais (,) (cosq,sin q ) 39

40 Rotate b q : ais (,) q q (,) 4

41 Rotate b q: ais (,) cosq q q 4243 sinq 4

42 Rotate b q : ais (,) ( sinq,cosq ) 42

43 Summar of Rotation b q (,) (cos q,sin q ) (,) ( sinq,cosq ) 43

44 Summar (Column Form) cosq sinq sinq cosq 44

45 Using Matri Notation cosq sinq -sinq cosq = cosq sinq cosq sinq -sinq cosq = -sinq cosq (Note that unit vectors simpl cop columns) 45

46 General Rotation b q Matri cosq -sinq cosq -sinq = sinq cosq sinq + cosq 46

47 Outline Toda s lecture Transformations and projective geometr 2D scales, shears, rotations, translations; homogeneous coordinates 47

48 Translations For a translation T, T(,)!= (,) but for an matri X, A(,) = (,) we get around this b going up to one higher dimension,, w 48

49 w = = 49

50 Translation in d + d = 5

51 Translation in d = + d 5

52 Homogeneous Coordinates = 52

53 Homogeneous Coordinates l l l =, for l 53

54 Homogeneous Coordinates For l, l l = = l l l l Homogeneous term affects overall scaling 54

55 Homogeneous Coordinates infinite number of points correspond to (,, ) constitute the whole line ( t,t,t) projected on plane w= hence projective geometr w ( t,t,t) (,,) w = 55

56 What does a 3D shear in do? w ( t, t, t) ( t, t, t) (,, ) (,, ) w = Translation in w= 56

57 Using Homogeneous Coord s Shear in Effects 3D translation in 2D We have used a linear transformation ( shear) in 3D to effect a nonlinear transformation ( translation) in 2D 57

58 Affine Transformations: Linear + Translation 58

59 Elementar Transformations Scale S ( l ), S ( l ) Rotate R( q ) Translate T ( d ), T ( d ) Shear Sh ( d ), Sh ( d ) 59

60 Refection about -ais = 6

61 Reflection about -ais (, ) (, ) (,) (,) 6

62 Reflection about -ais = - 62

63 Reflection about -ais (,) (, ) (, ) (, ) 63

64 Reflection Elementar Is? Can we effect reflection in a more elementar wa? in terms of scale, shear, rotation, translation? 64 64

65 Reflection is Scale (-)

66 Compound Transformations Build up compound transformations b concatenating elementar ones Use for complicated motion Use for complicated modeling 66

67 E: Advance clock hands ( a, b) 67

68 E: Advance clock hands ( a, b) 68

69 E: Advance clock hands ( a, b) 69

70 E: Advance clock hands ( a, b) 7

71 Clock Transformations Translate to Origin Move hand with rotation Move hand back to clock Do other hand 7

72 Clock Transformations T T s b = T ( a, b) R( t) T ( a, b) ( ) = T ( a, b) R 2 t T ( a, b) where t = 5 o ( hr = 3 o ) 72

73 Clock Transformations a cos( t) sin( t) a b sin( t) cos( t) b Translate Back Rotate About Origin Translate to Origin = 73

74 Outline Toda s lecture Transformations and projective geometr 2D scales, shears, rotations, translations; homogeneous coordinates 3D transformations 74

75 3D Transformations Scale S ( l ), S ( l ), S z ( l ) Rotate R ( q ), R ( q ), R z ( q ) Translate T ( d ), T ( d ), T z ( d ) Shear Sh ( d ), Sh ( d ), Sh z ( d ) 75

76 3D Scale in S ( l ) = l 76

77 77 3D 3D Scale Scale in in = = z z S l l

78 78 ) ( 3D 3D Scale Scale in in = = z z S l l l

79 79 ) ( 3D 3D Scale Scale z in in = = z z S z l l l

80 8 ) ( Overall (Uniform) Overall (Uniform) 3D 3D Scale Scale ( ) ( ) = = l l l z z S

81 8 Overall (Uniform) Overall (Uniform) Same in and 3D 3D Scale Scale, z: ( ) = z z z l l l l l l l

82 Positive Rotation 3D in? Sit at + end of given ais Look at Origin CCW Rotation is Positive direction 82 82

83 3D Positive Rotations z

84 3D Rotation about z-ais b q We have alread done this: Rz ( q ) = cosq sinq sinq cosq z 84

85 3D Rotation about -ais b q z (,,) q q (,,) 85

86 86 cos sin sin cos ) ( 3D 3D Rotation about Rotation about b -ais b ais = z R q q q q q q

87 3D Rotation about -ais b q z (,,) (,,) q q 87 87

88 88 cos sin sin cos ) ( = z R q q q q q q 3D 3D Rotation about Rotation about b -ais b ais

89 Elementar Transformations Scale Rotate Translate Shear (Reflect) S ( l ), ( l S ) R ( q ), ( q R ) T ( d ), T ( d ) Sh ( d ), Sh ( d Rf, Rf ) 89

90 Announcements S/U option Skip final project Shirle book CS 84 book this semester 9

91 HW for Tues Install Microsoft visual studio Do first 2 OpenGL tutorials from : Setting Up An OpenGL Window 2: Your First Polgon Add a second triangle to the scene somewhere Turn in a screen shot of output Or if ou get stuck, the error message Alternatel, show me an other OpenGL program ou have alread written (running on our current computer) 9

92 92

93 Acknowledgements Rich Riesenfeld) M adaptations of his slides, based on the book b Peter Shirle chapter 6 (2nd edition) Rahul Khardekar Ian Buck & Trond Hagen 93