简单变换 (Simple)
- 恒等变换 (Identity)
- 平移变换 (Translation)
- 旋转变换 (Rotation)
- 均匀缩放 (Isotropic/Uniform Scaling)
- 非均匀缩放 (Scaling)
- 镜像翻转 (Reflection)
- 剪切变换 (Shear)
- 透视变换 (Perspective)
变换的组合
刚体/欧式变换 (Rigid-Body/Euclidean)
-
恒等 (Identity) + 平移 (Translation) + 旋转 (Rotation)
-
变换前后保持距离 (distance) 和角度 (angle) 不变
相似变换 (Similitudes/Similarity)
-
刚体变换 (Rigid) + 均匀缩放 (Uniform Scaling)
-
只保持角度 (angle) 不变，不保持距离 (distance) 不变（有缩放）
线性变换 (Linear)
- 相似变换去掉平移，再加上另外 3 个
- 恒等 (Identity) + 旋转 (Rotation) + 均匀缩放 (Uniform Scaling) + 非均匀缩放 (Scaling) + 镜像翻转 (Reflection) + 剪切 (Shear)
- \(L(p+q)=L(p)+L(q)\)
- \(L(ap)=aL(p)\)
仿射变换 (Affine)
- 线性变换 (Linear) + 平移 (Translation)
投影变换 (Projective)
- 仿射变换 (Affine) + 透视变换 (Perspective)
| 组合 | Identity | Translation | Rotation | Uniform Scaling | Scaling | Reflection | Shear | Perspective |
|---|---|---|---|---|---|---|---|---|
| Rigid | ✅ | ✅ | ✅ | |||||
| Similarity | ✅ | ✅ | ✅ | ✅ | ||||
| Linear | ✅ | ✅ | ✅ | ✅ | ✅ | ✅ | ||
| Affine | ✅ | ✅ | ✅ | ✅ | ✅ | ✅ | ✅ | |
| Projective | ✅ | ✅ | ✅ | ✅ | ✅ | ✅ | ✅ | ✅ |