Alternative Parametrizations in Space and Manifolds

Points in various spaces $\mathbb{R}^n$ can be described using coordinates different from Cartesian ones. In this section, we present some useful parametrizations that will be used throughout the book.

We introduce the following definition:

Definizione 3   $S^{n}$ is the unit sphere in $\mathbb{R}^{n}$ such that:

$$S^{n} := \left\{ \mathbf{x} \in \mathbb{R}^{n} : \Vert \mathbf{x} \Vert^2 = 1 \right\}$$ (1.15)

Since a generic parametrization in $S^{n}$ will have $n+1$ components and only 1 constraint, it must possess $n$ degrees of freedom (DOF) by definition.

In case $n=0$, the "unit sphere" $S^{0} = \left\{ -1, +1 \right\}$ consists of only 2 points and therefore is not a connected manifold.

With $n=1$, the manifold is exactly the same as $SO(2)$ that is with the parametrization $S^{1} = \left\{ \begin{bmatrix} \cos \alpha \\ \sin \alpha \end{bmatrix} ; \alpha \in \mathbb{R} \right\}$.

The sphere $S^{2}$ (the surface of the sphere or a direction in $\mathbb{R}^3$) is instead a 2-manifold that lacks group structure. It can be parametrized by two parameters (for example, polar coordinates as we will see shortly) but will always exhibit singularities.