Conic Sections and Meet Intersections in Geometric Algebra

TL;DR

This paper explores conic sections within various geometric algebra frameworks, providing formulations and intersection calculations for lines, circles, spheres, and planes, including real and virtual intersections, with a focus on the conformal model.

Contribution

It offers a comprehensive overview of conic descriptions in geometric algebra and detailed intersection calculations in the conformal model, including virtual cases.

Findings

Formulations of conic sections in Euclidean, projective, and conformal algebras

Explicit calculations of intersections for lines, circles, spheres, and planes

Analysis of hyperbolic carriers in virtual intersections

Abstract

This paper first gives a brief overview over some interesting descriptions of conic sections, showing formulations in the three geometric algebras of Euclidean spaces, projective spaces, and the conformal model of Euclidean space. Second the conformal model descriptions of a subset of conic sections are listed in parametrizations specific for the use in the main part of the paper. In the third main part the meets of lines and circles, and of spheres and planes are calculated for all cases of real and virtual intersections. In the discussion special attention is on the hyperbolic carriers of the virtual intersections.