I am trying to split the space of $\mathrm{SO}(3)$ into spaces $(S_i)_{i\in\{1, \cdots, n\}}$ of rotations providing a regular paving of $\mathrm{SO}(3)$.
At least I would need that $\bigcup_{i=1}^n S_i = \mathrm{SO}(3)$ and $$ \forall i, \max_{(R_a, R_b) \in S_i \times S_i} m < d(R_a, R_b) < M $$ with:
- $m$ and $M$ two thresholds (ideally as close as possible from each other) to be adapted in function of $n$, and
- $d$ giving the angular distance between two attitudes.
I considered working with quaternions and generating points evenly spaced over $\mathbb{S}^3$ (that would be the centers of the $S_i$), for instance by considering regular 4-polytopes (like 24-cell) but I am not sure it is the most direct way to reach my goal... Furthermore this method is not very adaptable (I can only choose $n$ among the number of vertex of a few regular polytopes).
Ideally, I am looking for the following supplementary properties (independently):
- $\forall i,j:\; i\neq j \implies (S_i \cap S_j=\emptyset)$
- $m=M$
- $S_i$ can be easily indexed, i.e. given any attitude $R\in \mathrm{SO}(3)$ I can quickly find the (idealy unique) $i$ such that $R\in S_i$. I see/imagine this as the result of a stereographic projection well chosen that offers simple boundaries between the projections of $S_i$
- in a perfect word, the chosen $S_i$ would have some properties of symmetry regarding the octahedral group (I will denote $\mathcal{O} \leqslant \mathrm{SO(3)}$ the sub-group of the 24 rotations which coefficients of the associated rotation matrix belong to $\{-1,0,1\}$):
- $\forall S_i, \forall \rho \in \mathcal{O},\,\exists !\, S_j : \forall R \in S_i,\;(\rho \cdot R) \in S_j$
- (optionnaly) $SO(3)$ can be generated from a collection of connected $\frac{n}{24}$ subspaces among the $S_i$.
Thank you by advance if you can make any suggestion to guide my search ! I read some articles about charts on $\mathrm{SO}(3)$ but I did not manage to find a practical method to build a regular paving of SO(3) (or I did not understand I found it).