3D Transformation

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本章节首先介绍了基本的转换，根据数学规则将一个充分或表达式转换成另一个操作；另外从代数角度分析了我们常见的普通变换，此外还介绍了组合变换，从矩阵代数角度分析2D仿射变换，分析了这几种转换的矢量和标准。

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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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