The definition of index of maps f : P1 --> P2 applies to all maps fH : P1 --> P2 given by composing the rational normal curve with a linear projection from a (d-3)-plane H. A map fH with inflection points s1, s2, ... sn with index (ai, bi, ci) at point si has degree d-A, were A is the sum of the ai. As a curve of degree d-A, it has index (0, bi-ai, ci-ai) at point si.
Given a map fH : P1 --> P2, we may consider its dual curve, which is defined at all points s whose index has b=1 (and hence a=0) as the tangent line to the image of fH. That is, the line spanned by fH(s) and f'H(s). This gives a map from an open subset of P1 to the dual projective plane, which we identify with P2. By the Riemann extension theorem, this extends to a regular map defined at all points of P1.
Suppose that we have a map with index 0, b, c) at the point 0. Then there are coordinates for P2 in which the terms of lowest order are [1, sb, sc]. Thus its derivative has terms of lowest order [0, sb-1, sc-1]. The dual curve is given by the linear form which vanishes on these two vectors, and so its has terms of lowest order, namely [(b-c)sc, (c)sc-b, -b]. Thus the index of the dual curve at 0 is [0,c-b, c].
When the original curve has degree d and inflection points s1, s2, ..., sn with index (0, bi, ci) at point si, the sum of the degrees of these inflection points is 3(d-2), and so we have
| b1 + c1 - 3 + ... + bn + cn - 3 = 3(d - 2) |
| 3(d' - 2) | = | c1 - b1 + c1 - 3 + ... + cn - bn + cn - 3 |
| = | 2(c1 + b1 - 3) - 3(b1 - 1) + ... + 2(cn + bn - 3) - 3(bn - 1) | |
| = | 6(d - 2) - 3 (b1 - 1 + ... + bn - 1) |