The objective of this chapter is to obtain a complete list of conics (up to the relation of congruence) and explore the consequences of that listing. We start with an arbitrary conic Q and reduce the number of terms in it till we reach a ‘normal form’, a congruent conic given by a particularly simple formula. Section 15.1 is a key step in this process, rotating the axes till they are parallel to the coordinate axes. There are three main cases: Q has a unique centre, a line of centres, or no centre. In each case we will derive a list of ‘normal forms’. The net result is a listing into nine ‘classes’, each of which (with one exception) involves moduli. The question of distinguishing these classes, and ensuring there are no redundancies amongst the normal forms, provides the material for the next chapter.

The classification yields significant gains. The simplicity of the ‘normal form’ allows us to elucidate its geometry with relative ease. However, the relation of ‘congruence’ preserves all the desirable geometric features of a conic. Thus in principle we can access the geometry of any conic, without recourse to complex calculations. The first gain is the fact that any parabola, real ellipse, or hyperbola is constructible, so the interesting metric geometry of constructible conics can be extended to the three most important conic classes. Another geometric gain is that we can relate eigenvalues to axis lengths, leading to a practical technique for calculating axis lengths directly from equations.