Hexanions: 6D Space for Twists

Some of the popular representations for the motion of a rigid object are 4×4 real matrices or combinations of vector and quaternion [Ale02][DKL98]. A drawback of using 4 × 4 real matrices is their over-general purpose. A real matrix is capable of representing a far too rich set of transformations, for instance scaling and shear. Another drawback with matrices is the accumulation of numerical errors, which impose to renormalize the matrix once in a while. On the other hand, a drawback of using a combination of vector and quaternion is the separation of the motion in two components that have to be handled individually. This component separation can make implementation on computing devices unclear or unnecessary long. While there are surely other alternatives for representing numerically rigid motion, we propose here a new representation called hexanion. A hexanion is a compact and natural representation of the rigid transformations: translations, rotations and twists. The number of dimensions of hexanion space is six, which is also the degrees of freedom of rigid transformations. 1 Motivation The motivation behind the formalism of hexanions is the necessity to compute fractions of rigid transformation. For a transformation represented by a 4 × 4 real matrix, taking a fraction h can be done with the following: M = exp(h log M) (1) where exp M = ∑∞ k=0 M k! log M = − ∑∞ k=1 (I−M) k (2) M. Alexa et al. [Ale02] proposes to use numerical methods to achieve this in the general case. A matrix is however a poor representation for rigid transformations. A better choice is to use a hexanions, which can be interpreted as a shorthand for log M when M is a twist. A hexanion is a 6d element denoted with a pair of vectors: 〈~ ω, ~ m〉 (3) The relation beween a hexanion and the logarithm of a twist is: