Polar curve

It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas.

Each of these tangents touches C at one of the points of intersection of C and the first polar, and by Bézout's theorem there are at most n(n−1) of these.

This puts an upper bound of n(n−1) on the class of a curve of degree n. The class may be computed exactly by counting the number and type of singular points on C (see Plücker formula).

The p-th polar of a C for a natural number p is defined as ΔQpf(x, y, z) = 0.

By Bézout's theorem these curves have (n−1)2 points of intersection and these are the poles of L.[2] For a given point Q=(a, b, c), the polar conic is the locus of points P so that Q is on the second polar of P. In other words, the equation of the polar conic is The conic is degenerate if and only if the determinant of the Hessian of f, vanishes.

The elliptic curve E : 4 Y 2 Z = X 3 − XZ 2 in blue, and its polar curve ( E ) : 4 Y 2 = 2.7 X 2 − 2 XZ − 0.9Z 2 for the point Q = (0.9, 0) in red. The black lines show the tangents to E at the intersection points of E and its first polar with respect to Q meeting at Q .