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3D Transformation
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1.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals 3D Transformations CMSC 435/634

2.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Transformation Webster: The operation of changing one configuration or expression into another in accordance with a mathematical rule

3.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Transformation Webster: The operation of changing one configuration or expression into another in accordance with a mathematical rule

4.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Transformation Webster: The operation of changing one configuration or expression into another in accordance with a mathematical rule

5.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Transformation Webster: The operation of changing one configuration or expression into another in accordance with a mathematical rule

6.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin

7.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin

8.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin

9.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin • Transform is a vector to vector function • p = f (p)

10.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin • Transform is a vector to vector function • p = f (p) • Relativity: • From p point of view, object is transformed

11.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin • Transform is a vector to vector function • p = f (p) • Relativity: • From p point of view, object is transformed • From p point of view, coordinate system changes

12.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Using Transformation • Points on object represented as vector offset from origin • Transform is a vector to vector function • p = f (p) • Relativity: • From p point of view, object is transformed • From p point of view, coordinate system changes • Inverse transform, p = f −1 (p )

13.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters

14.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters • T (p)

15.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters • R(T (p))

16.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters • R(T (p)) = R ◦ T (p)

17.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters • R(T (p)) = R ◦ T (p) • R(p)

18.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters • R(T (p)) = R ◦ T (p) • T (R(p))

19.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Composing Transforms • Order matters • R(T (p)) = R ◦ T (p) • T (R(p)) = T ◦ R(p)

20.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order

21.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order • (R ◦ T )−1 (p )

22.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order • (R ◦ T )−1 (p ) = R −1 (p )

23.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order • (R ◦ T )−1 (p ) = T −1 (R −1 (p ))

24.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order • (R ◦ T )−1 (p ) = T −1 (R −1 (p )) • (T ◦ R)−1 (p )

25.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order • (R ◦ T )−1 (p ) = T −1 (R −1 (p )) • (T ◦ R)−1 (p ) = T −1 (p )

26.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Inverting Composed Transforms • Reverse order • (R ◦ T )−1 (p ) = T −1 (R −1 (p )) • (T ◦ R)−1 (p ) = R −1 (T −1 (p ))

27.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Translation • p =p+t

28.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Translation • p =p+t px   x  x  x p + tx   p t • p y  = p y  + t y  = p y + t y  pz pz tz pz + t z

29.Generic Transforms Common Transforms Composing Transforms Affine Transforms Vectors and Normals Translation • p =p+t px   x  x  x p + tx   p t • p y  = p y  + t y  = p y + t y  pz pz tz pz + t z • t says where p-space origin ends up (p = 0 + t)

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