Conic sections through five points - classical, projective, conformal
Proceedings of the International Symposium 2003 of Advanced Mechanical Engineering, Pukyong National University, Busan, Korea, 22-25 Nov. 2003, pp. 109-114 (2003)
Abstract: In the so-called conformal model of Euclidean space of geometric algebra, circles receive a very elegant description by the outer product of three general points of that circle, forming what is called a tri-vector. Because circles are a special kind of conic section, the question arises, whether in general some kind of third order outer product of five points on a conic section (or certain linear combinations) may be able to describe other types of conic sections as well. The main idea pursued in this paper is to follow up a formula of Grassmann for conic sections through five points and implement it in the conformal model. Grassmann obviously based his formula on Pascal's theorem. At the end we consider a simple linear combination of circle tri-vectors.