Chapter 14 - Conic Sections

Abstract

This chapter discusses conic sections. Conic sections are invariant under a more general type of map: the so-called projective maps. These maps are studied in projective geometry. Polynomial curves, bear a close relationship to affine geometry. Consequently, the de Casteljau algorithm makes use of ratios, which are the fundamental invariant of affine maps. Thus, the class of polynomial curves is invariant under affine transformations: an affine map maps a polynomial curve onto another polynomial curve. The fact that five points are sufficient to determine a conic is a consequence of the most fundamental theorem in the theory of conics. A large number of methods exist to construct conic sections from the given pieces of information, most of which is based on Pascal's theorem. In a projective environment, all conics are equivalent: projective maps map conics to conics. In affine geometry, conics fall into three classes: (1) hyperbolas, (2) parabolas, and (3) ellipses. Thus, ellipses are mapped to ellipses under affine maps, parabolas to parabolas, and hyperbolas to hyperbolas.