Hyperquaternions: An Efficient Mathematical Formalism for Geometry

Patrick Girard (1) , Patrick Clarysse (2, 3) , Romaric Pujol (4) , Robert Goutte (1) , Philippe Delachartre (5)

1 CREATIS - Centre de Recherche en Acquisition et Traitement de l'Image pour la Santé
2 MOTIVATE - Imagerie et modélisation Vasculaires, Thoraciques et Cérébrales
3 MYRIAD - Modeling & analysis for medical imaging and Diagnosis
4 INSA - Institut National des Sciences Appliquées
5 Imagerie Ultrasonore

Hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof), they constitute Clifford algebras endowed with an associative exterior product providing an efficient mathematical formalism for differential geometry. The paper presents a hyperquaternion formulation of pseudo-euclidean rotations and the Poincaré groups in n dimensions (via dual hyperquaternions). A canonical decomposition of these groups is developed as an extension of an euclidean formalism and illustrated by a 5D example. Potential applications include in particular, moving reference frames and machine learning.

Mots clés: Quaternions; Hyperquaternions; Pseudo-euclidean rotations; Poincaré groups; Canonical decomposition

Domaines: Géométrie algébrique [math.AG]

Patrick Girard, Patrick Clarysse, Romaric Pujol, Robert Goutte, Philippe Delachartre. Hyperquaternions: An Efficient Mathematical Formalism for Geometry. Geometric Science of Information. 4th International Conference (GSI 2019), Frank Nielsen; Frédéric Barbaresco, Aug 2019, Toulouse, France. pp.116-125, ⟨10.1007/978-3-030-26980-7_13⟩. ⟨hal-02276429⟩