Linearized Expressions of 3D Rotational Motion

Rotation motion in a three-dimensional physical world refers to an angular displacement of an object around a specific axis in $\mathbb{R}^3$. It is typically formulated as a non-linear and non-convex motion due to the nonlinearity and nonconvexity of $\mathbb{SO}(3)$. However, this paper proposes a new perspective that the 3D rotation motion can be expressed by a linear system without dropping any constraints and increasing any singularities. Moreover, two frequent cases, i.e., $\angle\left(\mathbf{R}\boldsymbol{x},\boldsymbol{y}\right)=0$ and $\angle\left(\mathbf{R}\boldsymbol{x},\boldsymbol{y}\right)=\frac{\pi}{2}$, in computer vision and robotics that can be expressed linearly are deeply discussed in this paper.