Triple Conformal Geometric Algebra for Cubic Plane Curves

Abstract

The triple conformal geometric algebra (TCGA) for the Euclidean ‐plane extends CGA as the product of 3 orthogonal CGAs and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3‐vectors that linearize the representation of nonlinear curves, and the entities are inner product null spaces with respect to all points on the represented curves. Each inner product null space entity also has a dual geometric outer product null space form. Orthogonal or conformal (angle preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line, or circle are possible. The TCGA defines commutator‐based differential operators in the coordinate directions that can be combined to yield a general n‐directional derivative.