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

Authors: Eckhard Hitzer, Dietmar Hildenbrand

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 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). Keywords: Clifford algebra, conformal geometric algebra, cubic curves, cubic surfaces, intersections

Comments: 11 Pages. accepted for M. Gavrilova et al (eds.), Proceedings of Workshop ENGAGE 2019 at CGI 2019 with Springer LNCS, April 2019, 1 table, corrections: 03+11 May 2019.

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Submission history

[v1] 2019-05-03 05:23:22
[v2] 2019-05-03 10:07:54
[v3] 2019-05-11 06:44:00

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