李群: SE(3)SE(3)SE(3)
T∈R4×4T \in \mathbb{R}^{4 \times 4}T∈R4×4
T=[Rt0T1]=[abcjdefkghil0001]T = \begin{bmatrix}
\mathbf{R}&\mathbf{t} \\
\mathbf{0}^T &1
\end{bmatrix} =
\begin{bmatrix}
a & b & c & j\\
d & e& f & k\\
g & h & i &l \\
0 & 0 &0 &1
\end{bmatrix}T=[R0Tt1]=⎣⎡adg0beh0cfi0jkl1⎦⎤
一个三维齐次坐标 PPP
P=[v1v2v31]P = \begin{bmatrix}
v_1 \\
v_2 \\
v_3 \\
1
\end{bmatrix} P=⎣⎡v1v2v31⎦⎤
TP=[av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l1]TP= \begin{bmatrix}
a v_1+ bv_2 + cv_3 + j\\
d v_1+ev_2 + fv_3 + k\\
g v_1+ hv_2 + i v_3+l \\
1
\end{bmatrix}TP=⎣⎡av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l1⎦⎤
TPun_homogeneous=[av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l]
TP_{un\_homogeneous}= \begin{bmatrix}
a v_1+ bv_2 + cv_3 + j\\
d v_1+ev_2 + fv_3 + k\\
g v_1+ hv_2 + i v_3+l \\
\end{bmatrix}
TPun_homogeneous=⎣⎡av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l⎦⎤