Top

Published in:

2019 | OriginalPaper | Chapter

Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra

Authors : Eckhard Hitzer, Dietmar Hildenbrand

Published in: Advances in Computer Graphics

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9, 7) over the real sixteen dimensional vector space \(\mathbb {R}^{9,7}\). These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19, 16).

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces

next chapter Non-parametric Realtime Rendering of Subspace Objects in Arbitrary Geometric Algebras

The use of the factor one half in \(\tfrac{1}{2} (x^2 \mathbf {e}_{\infty 1} + y^2 \mathbf {e}_{\infty 2})\) is taken over from the point definition in standard CGA [6], and has importance in preserving the inner product to distance relationship of CGA in (26).

The operation \((\mathbf {x} \wedge \mathbf {I}_{\infty }^\rhd ) \lfloor \mathbf {I}_{o}^\rhd \) combining outer product and contraction is typical for projection operations in geometric algebra. For example, in Cl(3, 0) the projection of multivector \(\mathbf {a}\) onto a blade \(\mathbf {b}\) is given by \((\mathbf {a}\wedge \mathbf {b})\lfloor \mathbf {b}^{-1}\). Since \(\mathbf {I}_{\infty }^\rhd \) is a product of null vectors and has no inverse, the projection operation is completed by contracting with \(\mathbf {I}_{o}^\rhd \) from the right, see (16).

This is a strategy similarly employed by Perwass for conics [16] and in [11], and for quadrics in [3, 12]. Treating the outer product of contact points (33) as the actual algebraic representation of the geometric object in question, was essential for the formulation of rotations, translations and scaling by means of versors in [12]. We intuitively expect that this may turn out to be similar in the current cubic CGA Cl(9, 7).

Abłamowicz, R.: Clifford algebra computations with maple. In: Baylis W.E. (eds), Clifford (Geometric) Algebras. Birkhäuser Boston (1996)CrossRef

Aragon-Camarasa, G., et al.: Clifford algebra with mathematica. In: Proceedings of the 29th International Conference on Applied Mathematics, Budapest (2015). Preprint: arXiv:0810.2412

Breuils, S., Nozick, V., Sugimoto, A., Hitzer, E.: Quadric conformal geometric algebra of \(\mathbb{R}^{9,6}\). Adv. Appl. Clifford Algebras 28(35), 1–16 (2018). https://doi.org/10.1007/s00006-018-0851-1MathSciNetCrossRefMATH

Breuils, S., Nozick, V., Fuchs, L.: GARAMON: geometric algebra library generator. In: Xambo-Descamps, S., et al. (eds.) Early Proceedings of the AGACSE 2018 Conference, 23–27 July 2018, Campinas, São Paulo, Brazil, pp. 97–106 (2018)

Buckley, A., Košir, T.: Determinantal representations of smooth cubic surfaces. Geom. Dedicata 125(1), 115–140 (2007). https://doi.org/10.1007/s10711-007-9144-xMathSciNetCrossRefMATH

Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann, Burlington (2007)

Easter, R.B., Hitzer, E.: Triple conformal geometric algebra for cubic plane curves. Math. Methods Appl. Sci. 41(11), 4088–4105 (2018). https://doi.org/10.1002/mma.4597. Preprint: http://vixra.org/pdf/1807.0091v1.pdfMathSciNetCrossRefMATH

Hildenbrand, D.: Introduction to Geometric Algebra Computing. CRC Press, Taylor & Francis Group, Boca Raton (2018)MATH

Hitzer, E.: The Creative Peace License. https://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/. Accessed 5 Apr 2019

10.

Hitzer, E., Tachibana, K., Buchholz, S., Yu, I.: Carrier method for the general evaluation and control of pose, molecular conformation, tracking, and the like. Adv. Appl. Clifford Algebras 19(2), 339–364 (2009). https://doi.org/10.1007/s00006-009-0160-9MathSciNetCrossRefMATH

11.

Hitzer, E., Sangwine, S. J.: Foundations of conic conformal geometric algebra and simplified versors for rotation, translation and scaling, to be published

12.

Hitzer, E.: Three-dimensional quadrics in conformal geometric algebras and their versor transformations. Adv. Appl. Clifford Algebras 29, 46 (2019). https://doi.org/10.1007/s00006-019-0964-1. Preprint: http://vixra.org/pdf/1902.0401v4.pdfMathSciNetCrossRef

13.

Hrdina, J., Navrat, A., Vasik, P.: Geometric algebra for conics. Adv. Appl. Clifford Algebras 28(66), 21 (2018). https://doi.org/10.1007/s00006-018-0879-2MathSciNetCrossRefMATH

14.

De Keninck, S.: ganja.js - geometric algebra for Javascript. https://github.com/enkimute/ganja.js. Accessed 03 May 2019

15.

Newstead, P.E.: Geometric invariant theory. In: Bradlow, S.B., et al. (eds.) Moduli Spaces and Vector Bundles, pp. 99–127. Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/CBO9781139107037.005. https://www.cimat.mx/Eventos/cvectorbundles/newsteadnotes.pdfCrossRef

16.

Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89068-3CrossRefMATH

17.

Sangwine, S.J., Hitzer, E.: Clifford multivector toolbox (for MATLAB). Adv. Appl. Clifford Algebras 27(1), 539–558 (2017). https://doi.org/10.1007/s00006-016-0666-x. Preprint: http://repository.essex.ac.uk/16434/1/authorfinal.pdfMathSciNetCrossRefMATH

Title: Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra
Authors: Eckhard Hitzer
Dietmar Hildenbrand
Publisher: Springer International Publishing
Book: Advances in Computer Graphics
Print ISBN: 978-3-030-22513-1

Electronic ISBN: 978-3-030-22514-8

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-22514-8_53

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"