Alternative Parameterizations in Spaces and Manifolds

Points in the various spaces $\mathbb{R}^n$ can be described using coordinates other than Cartesian coordinates. This section presents several useful parameterizations that will be used throughout the rest of the book.

We introduce the following definition:

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

$$S^{n} := \left\{ \mathbf{x} \in \mathbb{R}^{n} : \Vert \mathbf{x} \Vert^2 = 1 \right\}$$ (1.51)

Since a generic parameterization on $S^{n}$ has $n+1$ components and only one constraint, by definition it must have $n$ degrees of freedom (DOF).

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

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

The sphere $S^{2}$ (the surface of a sphere or a direction in $\mathbb{R}^3$), on the other hand, is a 2-manifold that does not have a group structure. It can be parameterized by two parameters (for example, polar coordinates, as we will see shortly), but it will always exhibit singularities.

The spheres $S^n$ and the rotation groups $SO(n)$ are examples of differentiable manifolds that will be used extensively to represent orientations and geometric poses.