Chapter IV - Conic Sections

Abstract

This chapter presents the introduction and history of conic sections. Incidence theorems about circles generalize to all conic sections. A set of points in the Euclidean plane is determined by the focus-directrix property only if it is either a Euclidean parabola, a Euclidean ellipse other than a circle, or a Euclidean hyperbola. Euclidean ellipses, Euclidean parabolas, and Euclidean hyperbolas are exactly the sets of all ordinary points on conic sections. It justifies taking a theorem about conic sections, choosing a position for the ideal line, and interpreting the special case that results as a theorem in the Euclidean plane about Euclidean ellipses, Euclidean parabolas, and Euclidean hyperbolas. It is also proved that there is a unique conic section through any five points, no three of which are collinear. As the conic sections to cross-ratios, one can relate them to nondegenerate quadratic equations.