Before beginning, let us define some standard mathematical notations.
| \(\mathbb{R}^n\) | The \(n\)-dimensional real number space. |
| \(O(n)\) | The set of all \(n \times n\) real square matrices \(M\) satisfying \(M^{tr}M=I\), where \(M^{tr}\) denotes the transpose of \(M\) and \(I\) is the identity matrix. It includes all transformations that preserve lengths and angles, satisfies all group axioms, and encompasses rotations, rotoinversions, reflections, and inversion operations in \(\mathbb{R}^n\) space. For rotations, the determinant is 1; for all others, it is -1. |
| \(SO(n)\) | The subset of \(O(n)\) with determinant equal to 1. That is, the set of all rotations in \(\mathbb{R}^n\) space. This is also a group, and it is a subgroup of \(O(n)\). The \(S\) stands for Special. |
| \(S^n\) | The \(n\)-dimensional unit sphere, defined as \(S^n = \{ x \in \mathbb{R}^{n+1}| \ \|x\|=1 \}\). It is the set of all points at unit distance from the origin in \(n+1\) dimensions. Note that the surface of an ordinary (3-dimensional) unit sphere is \(S^2\), not \(S^3\). The superscript refers to the dimension of the sphere’s surface, not the dimension of the space in which the sphere is embedded1. |