1 Dr. Scott Schaefer 3D Transformations. 2/149 Review – Vector Operations Dot Product.

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Presentation transcript:

1 Dr. Scott Schaefer 3D Transformations

2/149 Review – Vector Operations Dot Product

3/149 Review – Vector Operations Dot Product

4/149 Review – Vector Operations Dot Product

5/149 Review – Vector Operations Cross Product

6/149 Review – Vector Operations Cross Product

7/149 Review – Vector Operations Cross Product

8/149 Review – Vector Operations Cross Product

9/149 Review – Vector Operations Cross Product

10/149 Review – Vector Operations Cross Product

11/149 Review – Vector Operations Cross Product

12/149 Review – Vector Operations Cross Product

13/149 Review – Vector Operations Cross Product

14/149 Transformations Affine Transformations  Translation  Uniform Scaling  Non-uniform Scaling  Rotation  Mirror Image Projections  Orthogonal  Perspective

15/149 Translation

16/149 Uniform Scaling

17/149 Uniform Scaling

18/149 Uniform Scaling

19/149 Non-Uniform Scaling

20/149 Non-Uniform Scaling

21/149 Non-Uniform Scaling

22/149 Non-Uniform Scaling

23/149 Non-Uniform Scaling

24/149 Non-Uniform Scaling

25/149 Rotation

26/149 Rotation

27/149 Rotation

28/149 Rotation

29/149 Rotation

30/149 Rotation

31/149 Rotation

32/149 Rotation

33/149 Rotation

34/149 Mirror Image

35/149 Mirror Image

36/149 Mirror Image

37/149 Mirror Image

38/149 Mirror Image

39/149 Mirror Image Exactly the same as non-uniform scaling with !!!

40/149 Orthogonal Projection

41/149 Orthogonal Projection

42/149 Orthogonal Projection

43/149 Orthogonal Projection

44/149 Orthogonal Projection

45/149 Orthogonal Projection

46/149 Perspective Projection

47/149 Perspective Projection Plane:

48/149 Perspective Projection Plane: Line:

49/149 Perspective Projection Plane: Line:

50/149 Perspective Projection Plane: Line:

51/149 Perspective Projection Plane: Line:

52/149 Perspective Projection

53/149 Perspective Projection

54/149 Perspective Projection

55/149 Perspective Projection

56/149 Perspective Projection Perspective Projection is not defined for vectors!!!

57/149 Transformations as Matrices Exactly like 2D 4x4 matrices Requires coordinates…. 

58/149 Review – Vector Operations Dot Product

59/149 Review – Vector Operations Dot Product

60/149 Review – Vector Operations Cross Product

61/149 Review – Vector Operations Cross Product

62/149 Review – Vector Operations Cross Product

63/149 Review – Vector Operations

64/149 Review – Vector Operations

65/149 Review – Vector Operations

66/149 Translation

67/149 Translation

68/149 Uniform Scaling

69/149 Uniform Scaling

70/149 Uniform Scaling

71/149 Uniform Scaling

72/149 Uniform Scaling o = (0,0,0) T

73/149 Non-Uniform Scaling

74/149 Non-Uniform Scaling

75/149 Non-Uniform Scaling

76/149 Non-Uniform Scaling

77/149 Non-Uniform Scaling

78/149 Non-Uniform Scaling v = (1,0,0) T, o = (0,0,0) T

79/149 Rotation

80/149 Rotation

81/149 Rotation

82/149 Rotation

83/149 Rotation

84/149 Rotation

85/149 Rotation

86/149 Rotation

87/149 Rotation

88/149 Rotation Rotation about x

89/149 Rotation Rotation about y

90/149 Rotation Rotation about z

91/149 Mirror Image

92/149 Mirror Image

93/149 Mirror Image

94/149 Mirror Image

95/149 Mirror Image

96/149 Mirror Image

97/149 v = (0,0,1) T, o = (0,0,0) T Mirror Image

98/149 Orthogonal Projection

99/149 Orthogonal Projection

100/149 Orthogonal Projection

101/149 Orthogonal Projection

102/149 Orthogonal Projection

103/149 Orthogonal Projection

104/149 Orthogonal Projection v = (0,0,1) T, o = (0,0,0) T

105/149 Perspective Projection

106/149 Perspective Projection

107/149 Perspective Projection

108/149 Perspective Projection Perspective transformations are not Affine!!! Rational expression requires a slight modification (homogeneous coordinates)

109/149 Perspective Projection Perspective transformations are not Affine!!! Rational expression requires a slight modification (homogeneous coordinates)

110/149 Perspective Projection Perspective transformations are not Affine!!! Rational expression requires a slight modification (homogeneous coordinates)

111/149 Perspective Projection Perspective transformations are not Affine!!! Rational expression requires a slight modification (homogeneous coordinates) 3D location is

112/149 Perspective Projection

113/149 Perspective Projection

114/149 Perspective Projection e = (0,0,1) T, o = (0,0,0) T, v = (0,0,1) T

115/149 Hierarchical Animation Tree of transformation matrices Each node stores a local transformation with respect to its parent

116/149 Hierarchical Animation Tree of transformation matrices Each node stores a local transformation with respect to its parent

117/149 Hierarchical Animation Tree of transformation matrices Each node stores a local transformation with respect to its parent

118/149 Hierarchical Animation Tree of transformation matrices Each node stores a local transformation with respect to its parent

119/149 Skeletal Animation

120/149 Skeletal Animation

121/149 Skeletal Animation : Bone Transformation

122/149 Skeletal Animation : Skin Weights : Bone Transformation

123/149 Skeletal Animation : Skin Weights : Bone Transformation Image taken from [Wang et al. 2002]

124/149 Skeletal Animation Examples

125/149 Skeletal Animation Examples

126/149 Skeletal Animation Examples

127/149 Transformations in OpenGL Types of transformation matrices  View  Model  Projection  Viewport

128/149 Types of transformation matrices  View  Model  Projection  Viewport Transformations in OpenGL

129/149 Transformations in OpenGL Types of transformation matrices  View  Model  Projection  Viewport

130/149 Transformations in OpenGL Types of transformation matrices  View  Model  Projection  Viewport

131/149 Transformations in OpenGL Types of transformation matrices  View  Model  Projection  Viewport

132/149 OpenGL: Matrix Commands glTranslatef(x, y, z) glScalef(x, y, z) glRotatef(theta, vx, vy, vz) glLoadIdentity(void) glPushMatrix(void)/glPopMatrix(void)

133/149 OpenGL: Matrix Example glTranslatef(0, 1, 3)

134/149 OpenGL: Matrix Example glTranslatef(0, 1, 3) glRotatef(45, 0, 1, 0)

135/149 OpenGL: Matrix Example glTranslatef(0, 1, 3) glRotatef(45, 0, 1, 0) glScalef(1, 1, 2)

136/149 OpenGL: Matrix Example glTranslatef(0, 1, 3) glRotatef(45, 0, 1, 0) glScalef(1, 1, 2) OpenGL multiplies matrices in the opposite order of what you might expect

137/149 Coordinate Systems in OpenGL

138/149 Coordinate Systems in OpenGL

139/149 Coordinate Systems in OpenGL

140/149 Coordinate Systems in OpenGL

141/149 Coordinate Systems in OpenGL

142/149 Coordinate Systems in OpenGL

143/149 Coordinate Systems in OpenGL

144/149 Coordinate Systems in OpenGL

145/149 Coordinate Systems in OpenGL

146/149 Coordinate Systems in OpenGL 1.PushMatrix 2.Specify viewer using inverse of view transformation 3. PushMatrix 4. Position object with transformations in opposite order of what you would expect 5. PopMatrix 6. PopMatrix

147/149 Coordinate Systems in OpenGL 1.PushMatrix 2.Specify viewer using inverse of view transformation 3. PushMatrix 4. Position object with transformations in opposite order of what you would expect 5. PopMatrix 6. PopMatrix

148/149 Coordinate Systems in OpenGL 1.PushMatrix 2.Specify viewer using inverse of view transformation 3. PushMatrix 4. Position object with transformations in opposite order of what you would expect 5. PopMatrix 6. PopMatrix

149/149 OpenGL: Special Matrix Commands glMatrixMode(GL_MODELVIEW / GL_PROJECTION) gluLookAt(ex, ey, ez, cx, cy, cz, ux, uy, uz) gluPerspective(fov, aspect, near, far) glOrtho(left, right, bottom, top, near, far) glViewport(x, y, w, h)

150/149

151/149 Hierarchical Animation Typically represented as relative joint locations and rotations

152/149 Hierarchical Animation Typically represented as relative joint locations and rotations

153/149 Hierarchical Animation Typically represented as relative joint locations and rotations

154/149 Hierarchical Animation Typically represented as relative joint locations and rotations

155/149 Hierarchical Animation Typically represented as relative joint locations and rotations

156/149 Hierarchical Animation Typically represented as relative joint locations and rotations