E.M.S. Hitzer

**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.

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