Quadric Conformal Geometric Algebra of ${\mathbb {R}}^{9,6}$

Stéphane Breuils (1) , Vincent Nozick (2, 1) , Akihiro Sugimoto (3) , Eckhard Hitzer (4)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 JFLI - Japanese French Laboratory for Informatics
3 NII - National Institute of Informatics
4 ICU - International Christian University

Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-defined Conformal Geometric Algebra constructs lines, circles, planes, and spheres from control points just by using the outer product. There exist some Geometric Algebras to handle more complex objects such as quadric surfaces; however in this case, none of them is known to build quadric surfaces from control points. This paper presents a novel Geometric Algebra framework, the Geometric Algebra of R 9,6 , to deal with quadric surfaces where an arbitrary quadric surface is constructed by the mere wedge of nine points. The proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. Our proposed framework also provides the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface.

Mots clés: Geometric Algebra; Quadrics; Conformal Geometric Algebra; Clifford Algebra

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]; Traitement des images [eess.IV]

Stéphane Breuils, Vincent Nozick, Akihiro Sugimoto, Eckhard Hitzer. Quadric Conformal Geometric Algebra of ${\mathbb {R}}^{9,6}$. Advances in Applied Clifford Algebras, 2018, 28 (2), pp.35. ⟨10.1007/s00006-018-0851-1⟩. ⟨hal-01767230⟩