Discretization of SO(3) using recursive tesseract subdivision

Abstract

The group of rotations in three dimensions SO(3) plays a crucial role in applications ranging from robotics and aeronautics to computer graphics. Rotations have three degrees of freedom, but representing rotations is a nontrivial matter and different methods, such as Euler angles, quaternions, rotation matrices, and Rodrigues vectors are commonly used. Unfortunately, none of these representations allows easy discretization of orientations on evenly spaced grids. We present a novel discretization method that is based on a quaternion representation in conjunction with a recursive subdivision scheme of the four-dimensional hypercube, also known as the tesseract.