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

  • Stéphane Breuils
  • Vincent Nozick
  • Akihiro Sugimoto
  • Eckhard Hitzer
Article
Part of the following topical collections:
  1. T.C. : Geometric Algebra for Computing, Graphics and Engineering with Yu Zhaoyuan, Guest E-i-C

Abstract

Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-studied 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 \({\mathbb {R}}^{9,6}\), 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.

Keywords

Quadrics Geometric algebra Conformal geometric algebra Clifford algebra 

Mathematics Subject Classification

Primary 99Z99 Secondary 00A00 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire d’Informatique Gaspard-Monge, Equipe A3SIUMR 8049, Université Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  2. 2.CNRS JFLI, UMI 3527, National Institute of InformaticsTokyoJapan
  3. 3.National Institute of InformaticsTokyoJapan
  4. 4.International Christian UniversityTokyoJapan

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