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

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.