The geometry of the ellipse differs substantially from that of the parabola, since it has two axes of symmetry (whereas the parabola has just one) and is a central conic (whereas the parabola is not). Our first result is that all the lines passing through the centre meet the ellipse in two distinct points, distinguishing the ellipse visually from the hyperbola, and establishing the existence of exactly four vertices.

In Section 11.2 we take up the question of parametrization. Unlike parabolas, it is not possible to parametrize general ellipses by quadratic functions of a single variable. However ellipses can be parametrized in terms of rational functions, quotients of polynomial functions. Such rational parametrizations have interesting applications to other areas of mathematics. By way of illustration we indicate how to solve a problem of ancient Greek mathematics, that of listing right-angled triangles with integer sides.

The remainder of the chapter is devoted to focal properties of ellipses, in particular the interesting metric property that the sum of the distances from any point on the ellipse to the two foci is constant. The final section establishes a reflective property for ellipses, analogous to that for parabolas.

Axes and Vertices

Perhaps one of the most obvious properties of the circle is that every line through the centre cuts the circle twice. The ellipse should be thought of as a natural generalization of the circle, so one expects it to have the same property. Indeed that is the case.