A striking feature of a circle is that there is a point (the centre) which does not lie on the circle itself, yet is crucial to understanding the geometry of the curve. The concept of ‘centre’ is by no means unique to circles. Our first step is to introduce the idea for general conics: that provides the material for Section 5.1. However, general conics do not always have centres, presenting us with one crude way of distinguishing some conics from others. For that remark to be useful we need to have an efficient practical technique to find the centres of a conic, if any. That is the function of Section 5.2. These considerations enable us to distinguish three broad classes of conics, namely those having a unique centre, those having no centre, and those having a line of centres. And that will provide a basis for the classification of conics in Chapter 15.

The Concept of a Centre

We are used to thinking of the centre of a circle as the point equidistant from the points in its zero set. There is however another approach, capable of generalization. Any line through the centre meets the circle in two distinct points, and the centre is the midpoint of the resulting chord. That suggests how we might extend the concept to general conics. Let W = (u, v) be a fixed point.