Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment

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

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 XLIM-ASALI - Synthèse et analyse d'images
3 ICU - International Christian University
4 NII - National Institute of Informatics

We introduce the quadric conformal geometric algebra (QCGA) inside the algebra of R 9,6. In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of R 9,6 basis vectors.

Domaines: Géométrie algébrique [math.AG]

Stéphane Breuils, Laurent Fuchs, Eckhard Hitzer, Vincent Nozick, Akihiro Sugimoto. Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment. Advances in Applied Clifford Algebras, 2019, 29 (3), ⟨10.1007/s00006-019-0974-z⟩. ⟨hal-02169419⟩