Find Rotation Transformation Between Rotation Matrices
In the field of computer graphics and robotics, understanding the transformation between rotation matrices is crucial for various applications, such as camera calibration, motion planning, and augmented reality. A rotation matrix represents a rotation in three-dimensional space and is widely used in computer graphics and related fields. This article aims to discuss the process of finding the rotation transformation between two rotation matrices, which is essential for various computational tasks.
Rotation matrices are orthogonal matrices with determinant 1, representing a rotation in three-dimensional space. The general form of a rotation matrix is given by:
\[ R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
where \(\theta\) is the rotation angle around the \(z\)-axis. The rotation matrices can be transformed to represent rotations around the \(x\)- and \(y\)-axes using similar formulas.
The process of finding the rotation transformation between two rotation matrices involves determining the rotation angle and axis between the two matrices. One common method to achieve this is by using the Rodrigues’ rotation formula, which provides a way to compute the rotation matrix from a rotation axis and angle.
To find the rotation transformation between two rotation matrices \(R_1\) and \(R_2\), follow these steps:
1. Compute the difference between the two rotation matrices: \(R_{diff} = R_2 \cdot R_1^{-1}\).
2. Calculate the angle of rotation: \(\theta = \arccos\left(\frac{\text{trace}(R_{diff}) – 1}{2}\right)\).
3. Compute the rotation axis: \(u = \frac{R_{diff} \cdot \text{cross}(R_{diff}[0], R_{diff}[1])}{\text{norm}(R_{diff} \cdot \text{cross}(R_{diff}[0], R_{diff}[1]))}\).
4. Apply the Rodrigues’ rotation formula to obtain the rotation matrix \(R_{transformation}\) that represents the transformation between \(R_1\) and \(R_2\).
By following these steps, you can find the rotation transformation between two rotation matrices, which can be used in various applications such as computer graphics, robotics, and motion planning. It is important to note that this method assumes that the rotation matrices are orthonormal and represent rotations in three-dimensional space.
