Properties of the Rotation Matrix

The rotation matrix will often be referred to in the text, for the sake of brevity, as an array in the C programming language:

$$\mathbf{R} = \begin{bmatrix} r_{0} & r_{1} & r_{2} \\ r_{3} & r_{4} & r_{5} \\ r_{6} & r_{7} & r_{8} \end{bmatrix}$$

The rotation matrix is a highly overparameterized matrix: its 9 linearly independent parameters are, in fact, generated by 3 variables in a nonlinear manner (see appendix).

Without explicitly stating the angles from which the matrix is generated, it is possible to provide some additional constraints. The rotation matrix has the property of not altering distances, being orthonormal and $\det(\mathbf{R})=1$. Each row and each column must have unit length, and every row and every column are orthonormal to each other, as they form orthonormal bases of the space. Therefore, knowing two row or column vectors of the matrix $\mathbf{r}_1$ and $\mathbf{r}_2$, it is possible to determine the third basis vector as the cross product of the previous two:

$$\mathbf{r}_3 = \mathbf{r}_1 \times \mathbf{r}_2$$ (8.19)

Similarly, the dot product between two row vectors or two column vectors must yield a zero value, as they are orthogonal to each other. Under these constraints, there exist two exact solutions, one of which is:

$$\mathbf{R} = \begin{bmatrix} r_{0} & r_{1} & (1 - r_{0}^2 - ... ... + r_{1}^2 + r_{3}^2 + r_{4}^2 - 1)^{\frac{1}{2}} \end{bmatrix}$$ (8.20)

where $s = sgn(r_{1}r_{4} + r_{2}r_{5})$, while the other solution has exactly the opposite signs. Knowing a submatrix $2 \times 2$, it is possible to derive the other elements of the matrix itself up to a sign, always based on the fact that each row and column have unit norm.