A General Homogeneous Matrix Formulation to 3D Rotation Geometric Transformations
This work addresses a specific mathematical formulation gap in 3D geometry for applications in computer graphics, robotics, or vision, but appears incremental as it builds on known concepts like homogeneous matrices and Euler-Rodrigues formulas.
The authors tackled the problem of 3D rotation transformations when the rotation axis does not pass through the coordinate origin, proposing a general homogeneous matrix formulation that bridges a gap between applications and existing definitions. They presented formulas similar to the Euler-Rodrigues formula, with the matrix-vector form being particularly suited for numerical applications where gimbal lock is a concern.
We present algebraic projective geometry definitions of 3D rotations so as to bridge a small gap between the applications and the definitions of 3D rotations in homogeneous matrix form. A general homogeneous matrix formulation to 3D rotation geometric transformations is proposed which suits for the cases when the rotation axis is unnecessarily through the coordinate system origin given their rotation axes and rotation angles. General three-dimensional rotation formula~\eqref{eqn:3D homogeneous roation} and~\eqref{eqn:3D rotation matrix vector Euclidean} similar to the Euler-Rodrigues formula were presented. The matrix-vector form of 3D rotation in Euclidean space is especially suited for numerical applications where gimbal lock is a concern.}