Conics and Quadric Surfaces

Abstract

If we think of lines, planes and general affine subspaces as sets of points satisfying a linear equation then circles and spheres are examples of sets of points which satisfy a quadratic equation. The solutions to a quadratic equation in the plane are calledconic sectionsor conics for short. These were known to the ancient Greeks and were given this name because they can be thought of as the intersection of a plane with a circular cone. This is the definition we shall start with and we shall end with their definition in terms of a focus and directrix. The latter, introduces us to all sorts of pretty properties of a conic. The focus, as the name suggests, involves rays of light reflected by the conic curve with important practical consequences.