Conics, Their Pencils and Intersections in Geometric Algebra

This paper presents an approach for extracting points from conic intersections by using the concept of pencils. This method is based on QC2GA-the two-dimensional version of QCGA (Quadric Conformal Geometric Algebra)-that is demonstrated to be equivalent to GAC (Geometric Algebra for Conics). A new interpretation of QC2GA and its objects based on pencils of conics and point space elements is presented, enabling the creation, constraining, and exploitation of pencils of conics. A Geometric Algebra method for computing the discriminants and center point of a conic will also be presented, enabling the proposition of an algorithm for extracting points from a conic intersection object.