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On Specific Conic Intersections in GAC and Symbolic Calculations in GAALOPWeb

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A Correction to this article was published on 10 January 2022

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Abstract

We describe a possibility for geometric calculation of specific conics’ intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected to intersections of conics in a general mutual position. Then we show how symbolic operations may be calculated directly in GAALOPWeb software, that the basis coefficients may be read off in the appropriate basis and, moreover, the result may be immediately and truly visualized. We compare the functionality with Maple package Clifford.

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Notes

  1. Since GAC is able to handle rotations correctly compared to other algebras this is presented as an example in this chapter.

  2. The GAALOPWeb visualization is based on the ganja tool of Steven de Keninck as described in [7].

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Institute of Mathematics, Brno University of Technology, Brno, Czech Republic

    Roman Byrtus, Anna Derevianko & Petr Vašík

  2. Techische Universität Darmstadt, Darmstadt, Germany

    Dietmar Hildenbrand & Christian Steinmetz

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  1. Roman Byrtus
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  2. Anna Derevianko
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  3. Petr Vašík
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Correspondence to Petr Vašík.

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Roman Byrtus, Anna Derevianko and Petr Vašík were supported by a grant no. FSI-S-20-6187.

This article is part of the ENGAGE 2020 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Werner Benger, Dietmar Hildenbrand, Eckhard Hitzer, and George Papagiannakis.

The original version of this article was revised to update the formula on page 9 and some of the reference corrections.

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Byrtus, R., Derevianko, A., Vašík, P. et al. On Specific Conic Intersections in GAC and Symbolic Calculations in GAALOPWeb. Adv. Appl. Clifford Algebras 32, 2 (2022). https://doi.org/10.1007/s00006-021-01182-z

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  • DOI: https://doi.org/10.1007/s00006-021-01182-z

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