Nonsingular Efficient Modeling of Rotations in 3-space using three components

TL;DR

This paper presents a novel, singularity-free affine patch representation of 3D rotations that is more compact and computationally efficient than quaternions, with advantages in integration and state space properties.

Contribution

The paper introduces a new affine patch-based rotation representation in 3D space that avoids singularities and simplifies computations compared to existing methods.

Findings

The affine patch representation is more compact than quaternions.

It encompasses the entire rotation group without singularities.

Differential equations for angular velocity integration are derived with advantages over quaternion methods.

Abstract

This article introduces yet another representation of rotations in 3-space. The rotations form a 3-dimensional projective space, which fact has not been exploited in Computer Science. We use the four affine patches of this projective space to parametrize the rotations. This affine patch representation is more compact than quaternions (which require 4 components for calculations), encompasses the entire rotation group without singularities (unlike the Euler angles and rotation vector approaches), and requires only ratios of linear or quadratic polynomials for basic computations (unlike the Euler angles and rotation vector approaches which require transcendental functions). As an example, we derive the differential equation for the integration of angular velocity using this affine patch representation of rotations. We remark that the complexity of this equation is the same as the corresponding quaternion equation, but has advantages over the quaternion approach e.g. renormalization to unit length is not required, and state space has no dead directions.