Before proceeding, we define some standard mathematical notation used throughout this section.
| \(\mathbb{R}^n\) | 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. This group contains all transformations in \(\mathbb{R}^n\) that preserve lengths and angles, including rotations, rotoinversions, reflections, and inversions. The determinant equals 1 for proper rotations and −1 for the rest. |
| \(SO(n)\) | The subset of \(O(n)\) consisting of matrices with determinant 1; i.e., all proper rotations in \(\mathbb{R}^n\). This is also a group and a subgroup of \(O(n)\). The \(S\) stands for Special. |
| \(S^n\) | The \(n\)-dimensional unit sphere, defined as \(S^n = \{ x \in \mathbb{R}^{n+1}\mid \|x\|=1 \}\). It is the set of all points at unit distance from the origin in \((n+1)\)-dimensional space. Note that the surface of an ordinary (three-dimensional) unit ball is \(S^2\), not \(S^3\). The superscript denotes the dimension of the surface, not of the ambient space1. |
- \(S^1\) is a circle, \(S^2\) is a spherical surface, and \(S^3\) is a solid ball. ↩︎