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)