In this chapter we take a first look at more general conics than circles, before launching ourselves into more detailed considerations. One of our long-term objectives will be to separate out general conics into a small number of types, distinguished by their underlying geometry. We start by introducing the reader to the ‘standard’ conics which will play a dominant role in this text. They are not simply examples of conics: they turn out to be models of the physically most important conics, in a sense made precise in Chapter 15. The qualitative form of their zero sets can be determined by looking carefully at the way in which they intersect the pencils of horizontal and vertical lines, and offers insight into the computer generated illustrations. Like lines and circles the ‘standard’ conics admit natural parametrizations, of practical value in elucidating their geometry.

We will need simple and effective means for distinguishing one type of conic from another. As a first step in this direction we introduce three easily calculated ‘invariants’ of a general conic, namely the trace invariant τ, the delta invariant δ, and the discriminant Δ. All three are easily calculated expressions in the coefficients, from which we can read off useful geometric information. However, their true significance does not appear till the final chapter, where it is shown that they are ‘invariant’ in a strictly defined sense.