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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

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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).

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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).

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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

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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

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