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

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).

[1]  Stephen J. Sangwine,et al.  Clifford Multivector Toolbox (for MATLAB) , 2017 .

[2]  Dietmar Hildenbrand.  Introduction to Geometric Algebra Computing , 2018 .

[3]  Jaroslav Hrdina,et al.  Geometric Algebra for Conics , 2018, Advances in Applied Clifford Algebras.

[4]  D. Mumford,et al.  Geometric Invariant Theory , 2011 .

[5]  Eckhard Hitzer,et al.  Triple Conformal Geometric Algebra for Cubic Plane Curves , 2018 .

[6]  G. Aragon-Gonzalez,et al.  Clifford algebra with mathematica , 2008, 0810.2412.

[7]  Stephen Mann,et al.  Geometric algebra for computer science - an object-oriented approach to geometry , 2007, The Morgan Kaufmann series in computer graphics.

[8]  P. Newstead.  Moduli Spaces and Vector Bundles: Geometric Invariant Theory , 2009 .

[9]  P. E. Newstead.  Moduli Spaces and Vector Bundles: Geometric Invariant Theory , 2009 .

[10]  Eckhard Hitzer.  Three-Dimensional Quadrics in Conformal Geometric Algebras and Their Versor Transformations , 2019, Advances in Applied Clifford Algebras.

[11]  Kanta Tachibana,et al.  Carrier Method for the General Evaluation and Control of Pose, Molecular Conformation, Tracking, and the Like , 2009 .

[12]  Stephen J. Sangwine,et al.  Foundations of Conic Conformal Geometric Algebra and Compact Versors for Rotation, Translation and Scaling , 2019, Advances in Applied Clifford Algebras.

[13]  Akihiro Sugimoto,et al.  Quadric Conformal Geometric Algebra of $${\mathbb {R}}^{9,6}$$R9,6 , 2018 .

[14]  Tomaž Košir,et al.  Determinantal representations of smooth cubic surfaces , 2006 .

[15]  Christian Perwass,et al.  Geometric Algebra with Applications in Engineering , 2008, Geometry and Computing.

[16]  I. Todorov,et al.  Octonions, Exceptional Jordan Algebra and The Role of The Group $$F_4$$F4 in Particle Physics , 2018, Advances in Applied Clifford Algebras.

[17]  Rafał Abłamowicz.  Clifford Algebra Computations with Maple , 1996 .