Matrices de rotaciones, simetr\'{\i}as y roto-simetr\'{\i}as

TL;DR

This paper explicitly derives the orthogonal matrices representing rotations and symmetries in three-dimensional space, exploring their properties and relationships, including commutativity, with applications to geometric transformations.

Contribution

It provides explicit formulas for rotation and symmetry matrices in 3D, clarifies their dependence on parameters, and demonstrates their commutative property.

Findings

Matrices depend on vector components and rotation angle

Rotation matrix commutes with symmetry matrix

Explicit formulas for matrices are derived

Abstract

In this note we find the orthogonal matrices $R,S\in M_3(\mathbb{R})$ corresponding to the clockwise rotation $r$ in $\mathbb{R}^3$ around the axis generated by a unit vector $u=(a,b,c)^t$ through an angle $\alpha\in [0,2\pi)$, and to the symmetry $s$ in $\mathbb{R}^3$ on the plane perpendicular to $u$. Matrix $S$ depends on $a,b,c$ and matrix $R$ depends on $a,b,c, \cos \alpha$ and $\sin \alpha$. We show $SR=RS$. The matrix $R$ is due to Alperin.