Is there any definition for homogeneous rotations?

Question

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin.

Since homogeneous coordinates means algebraic representation in projective space where the concept distance becomes invalid and undefined, I am not sure whether there is any algebraic definition to homogeneous matrices of rotations in any textbook or literature?

So that I can identify the following homogeneous matrix as a rotation or not per such a definition:

$$\left( \begin{array}{cccc} 50+10 \sqrt{3} & 15 \sqrt{3}-\sqrt{35}-30 & 5 \sqrt{3}+3 \sqrt{35}-10 & 40-20 \sqrt{3}+4 \sqrt{35} \\ 15 \sqrt{3}+\sqrt{35}-30 & 18+26 \sqrt{3} & 6-3 \sqrt{3}+5 \sqrt{35} & 88-44 \sqrt{3}+4 \sqrt{35} \\ 5 \sqrt{3}-3 \sqrt{35} -10 & 6-3 \sqrt{3}-5 \sqrt{35} & 2+34 \sqrt{3} & 32 \sqrt{3}+8 \sqrt{35}-64 \\ 0 & 0 & 0 & 70 \\ \end{array} \right)$$

Answer 1

For such transform which don't actually project, i.e. which have the form $$ T = \begin{pmatrix} L_T & c_T \\ 0\,0 & t_T \end{pmatrix} \quad \text{where } L_T \in \mathbb{R}^{3\times 3},\, c_T \in \mathbb{R}^{3\times 1},\, t \neq 0 $$ you can simply check if $L_T$ is the matrix of a linear rotation in $\mathbb{R}^3$.

To verify that $L_T$ is indeed a rotation matrix, and find the angle and axis, you can use (one or more) of the following facts

1. $L_T$ is a rotation exactly if $\det L_T = 1$ and $L_T^* = L_T^{-1}$ (where $^*$ means tranposed, i.e. the transposed of $L_T$ is the inverse of $L_T$).

2. If $L_T$ is a rotation in $\mathbb{R}^3$ by angle $\varphi$ around the axis $a$, it has a real eigenvalue $1$ and complex eigenvalues $e^{i\phi},e^{-i\phi}$. The axis $a$ is an eigenvector corresponding to the real eigenvalue $1$, i.e. you can find it by solving $(L_T - I)x = 0$.

3. If $L_T$ is a rotation in $\mathbb{R}^3$ by angle $\varphi$, the trace (i.e. the sum of diagonal elements) of $L_T$ is $1 + 2\cos \varphi$.

Note that axis in the above refers to the axis of $L_T$, not the axis of $T$. To find the axis of $T$, you can e.g. find two independent solutions $x_1,x_2$ of $$ Tx = x \text{,} $$ i.e. two eigenvalues of $T$ with $x_1 \neq \lambda x_2$ for all $\lambda \in \mathbb{R}$.

Answer 2

I notice in this article A General Homogeneous Matrix Formulation to 3D Rotation Geometric Transformations there is a new definition on 3D rotations in exactly homogeneous matrices.

The definition is based on reflections, i.e., a rotation is the composition transformation of two reflection. This is actually a definition in the classical approach. However, the only difference might be, the reflection was defined in a completely different way by the authors in their own cited articles.

Hope this helps.