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)