齐次矩阵包含平移和旋转
第一个矩阵旋转矩阵
A = [ R 1 0 0 1 ] A=\left[\begin{matrix}R_{1} & 0\\0 & 1\end{matrix}\right] A=[R1001]
第一个平移矩阵
B = [ 1 T 1 0 1 ] B=\left[\begin{matrix}1 & T_{1}\\0 & 1\end{matrix}\right] B=[10T11]
C = [ R 2 0 0 1 ] C=\left[\begin{matrix}R_{2} & 0\\0 & 1\end{matrix}\right] C=[R2001]
两个矩阵的乘积
A B = [ R 1 R 1 T 1 0 1 ] AB=\left[\begin{matrix}R_{1} & R_{1} T_{1}\\0 & 1\end{matrix}\right] AB=[R10R1T11]
B A = [ R 1 T 1 0 1 ] BA=\left[\begin{matrix}R_{1} & T_{1}\\0 & 1\end{matrix}\right] BA=[R10T11]
B A B C = [ R 1 R 2 R 1 T 1 + T 1 0 1 ] BABC=\left[\begin{matrix}R_{1} R_{2} & R_{1} T_{1} + T_{1}\\0 & 1\end{matrix}\right] BABC=[R1R20R1T1+T11]
( B A B C ) − 1 = [ 1 R 1 R 2 − T 1 R 2 − T 1 R 1 R 2 0 1 ] (BABC)^{-1}=\left[\begin{matrix}\frac{1}{R_{1} R_{2}} & - \frac{T_{1}}{R_{2}} - \frac{T_{1}}{R_{1} R_{2}}\\0 & 1\end{matrix}\right] (BABC)−1=[R1R210−R2T1−R1R2T11]
我们可以看到，对于齐次矩阵中包含了旋转操作。取前三行和前三列就是旋转矩阵。