$$ \boldsymbol{A} = \begin{bmatrix} \boldsymbol{R} & \boldsymbol{T} \ \boldsymbol{0} & 1 \ \end{bmatrix} $$
时间: 2024-02-04 15:56:47 浏览: 22
& \boldsymbol{t} \\ \boldsymbol{0}^T & 1 \end{bmatrix} $$
where $\boldsymbol{R}$ is a $3\times 3$ rotation matrix and $\boldsymbol{t}$ is a $3\times 1$ translation vector. The matrix $\boldsymbol{A}$ represents a rigid body transformation in 3D, which can be thought of as a combination of rotation and translation.
To apply this transformation to a 3D point $\boldsymbol{p} = [x, y, z]^T$, we can use homogeneous coordinates by appending a 1 to the vector: $\boldsymbol{p}_h = [x, y, z, 1]^T$. Then, the transformed point $\boldsymbol{p}'$ can be obtained as:
$$ \boldsymbol{p}'_h = \boldsymbol{A} \boldsymbol{p}_h $$
After the transformation, we can convert the resulting homogeneous vector back to Cartesian coordinates by dividing by its last component:
$$ \boldsymbol{p}' = \frac{1}{w} \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} $$
where $w$ is the last component of $\boldsymbol{p}'_h$.
Note that applying a rigid body transformation to a point does not change the distance between any two points, i.e., it preserves distances. This property makes rigid body transformations useful in many applications, such as robotics, computer graphics, and computer vision.
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