In this chapter we will be concerned with the question of when two curves are to be regarded as the ‘same’. The first thing to be clear about is that there is no absolute answer – it all depends on which aspects of curves you are studying. We will introduce the group of ‘affine mappings’, leading to a natural relation of ‘affine equivalence’ on the curves of given degree d. The remainder of the chapter will be concerned with concepts which are invariant under affine mappings, so can be used in principle to distinguish curves. The first of these is the concept of degree, which has a basic geometrical interpretation, and the second is the interesting concept of a ‘centre’, inspired by the familiar case of a circle.

In order to motivate the definitions of Section 4.1, it may be helpful to start with what is possibly a more familiar notion, at least in the context of studying the standard conies of elementary geometry. We keep to the familiar Euclidean plane ℝ2. (This is one of the very few points in our development where we will refer to the Euclidean structure on ℝ2, but only for motivational purposes.) When dealing with standard conies, one thinks of two curves as being ‘rigidly’ equivalent when the one can be superimposed exactly on the other. You think of the curves as being in different (rectangular) coordinate systems, say one curve in the ‘standard’ (x, y)-plane, and the other in the (X, Y)-plane.