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

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Abstract

We introduce the quadric conformal geometric algebra inside the algebra of \({\mathbb {R}}^{9,6}\). 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 \({\mathbb {R}}^{9,6}\) basis vectors.

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Notes

  1. Git clone https://git.renater.fr/garamon.git.

  2. Note that the product symbols \(\rfloor \) and \(\lfloor \) express left- and right contraction, respectively.

  3. Git clone https://git.renater.fr/garamon.git.

  4. E. Hitzer, Creative Peace License, https://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/.

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Acknowledgements

We do thank the organizers of the international conference AGACSE 2018 for the inspiring event held in the summer of 2018 in Campinas, Brazil, that greatly facilitated our collaboration. EH wants to thank God Soli Deo Gloria, and invite all readers of this work to take the Creative Peace LicenseFootnote 4 into consideration, when applying this research.

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Authors and Affiliations

  1. Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI, UMR 8049, Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France

    Stéphane Breuils & Vincent Nozick

  2. Laboratoire XLIM-ASALI, UMR CNRS 7252, Université de Poitiers, Poitiers, France

    Laurent Fuchs

  3. International Christian University, Tokyo, 181-8585, Japan

    Eckhard Hitzer

  4. National Institute of Informatics, Tokyo, 101-8430, Japan

    Akihiro Sugimoto

Authors
  1. Stéphane Breuils
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  2. Laurent Fuchs
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  3. Eckhard Hitzer
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  4. Vincent Nozick
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  5. Akihiro Sugimoto
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Correspondence to Stéphane Breuils.

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This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECC-UNICAMP, Campinas, Brazil, edited by Sebastià Xambó-Descamps and Carlile Lavor.

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Breuils, S., Fuchs, L., Hitzer, E. et al. Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment. Adv. Appl. Clifford Algebras 29, 57 (2019). https://doi.org/10.1007/s00006-019-0974-z

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