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).
Soli Deo Gloria.
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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).
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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).
References
Abłamowicz, R.: Clifford algebra computations with maple. In: Baylis W.E. (eds), Clifford (Geometric) Algebras. Birkhäuser Boston (1996)
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/s00006-018-0851-1
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-x
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.pdf
Hildenbrand, D.: Introduction to Geometric Algebra Computing. CRC Press, Taylor & Francis Group, Boca Raton (2018)
Hitzer, E.: The Creative Peace License. https://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/. Accessed 5 Apr 2019
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/s00006-009-0160-9
Hitzer, E., Sangwine, S. J.: Foundations of conic conformal geometric algebra and simplified versors for rotation, translation and scaling, to be published
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.pdf
Hrdina, J., Navrat, A., Vasik, P.: Geometric algebra for conics. Adv. Appl. Clifford Algebras 28(66), 21 (2018). https://doi.org/10.1007/s00006-018-0879-2
De Keninck, S.: ganja.js - geometric algebra for Javascript. https://github.com/enkimute/ganja.js. Accessed 03 May 2019
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.pdf
Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89068-3
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.pdf
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Hitzer, E., Hildenbrand, D. (2019). Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra. In: Gavrilova, M., Chang, J., Thalmann, N., Hitzer, E., Ishikawa, H. (eds) Advances in Computer Graphics. CGI 2019. Lecture Notes in Computer Science(), vol 11542. Springer, Cham. https://doi.org/10.1007/978-3-030-22514-8_53
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