The concept of ‘tangent’, developed in Chapter 7 for affine curves, can be extended to projective curves, via the process of taking affine views. The result is a number of geometric insights. For instance, for conies it throws light (surprisingly) on the affine concept of a ‘centre’, and (even more surprisingly) on the metric concept of a ‘focus’. And for general affine curves, it enables us to understand better the common feature of ‘asymptotes’.

Tangents to Projective Curves

Tangent lines to projective curves are defined by analogy with the affine case. Let P be a point of multiplicity m on a projective curve F in PK2. Then I(P,F,L)≥m for every line L through P. We say that a line L is tangent to F at P, or a tangent line to F at P, when I(P,F,L) > m. The concept of tangency is invariant under projective maps in the following sense. Suppose that under a projective map P, F, L correspond to P′, F′, L′, then L is tangent to F at P if and only if L′ is tangent to F′ at P′. That follows immediately from the fact (Lemma 11.7) that intersection numbers (of curves with lines) are invariant under projective maps. Generally, we say that two curves F, G are tangent at the point P when they have a common tangent at P. In view of the above remarks the general concept of tangency is invariant under projective changes of coordinates.