Polar curve explained

In algebraic geometry, the first polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point of C whose tangent line passes through Q. It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas.

Definition

Let C be defined in homogeneous coordinates by f(x, y, z) = 0 where f is a homogeneous polynomial of degree n, and let the homogeneous coordinates of Q be (a, b, c). Define the operator

\Delta _{Q} = a{\partial\over\partial x} + b{\partial\over\partial y} + c{\partial\over\partial z}. Then Δ_Qf is a homogeneous polynomial of degree n−1 and Δ_Qf(x, y, z) = 0 defines a curve of degree n−1 called the first polar of C with respect of Q.

If P=(p, q, r) is a non-singular point on the curve C then the equation of the tangent at P is

x{\partial f\over\partial x}(p, q, r) + y{\partial f\over\partial y}(p, q, r) + z{\partial f\over\partial z}(p, q, r) =0 . In particular, P is on the intersection of C and its first polar with respect to Q if and only if Q is on the tangent to C at P. For a double point of C, the partial derivatives of f are all 0 so the first polar contains these points as well.

Class of a curve

The class of C may be defined as the number of tangents that may be drawn to C from a point not on C (counting multiplicities and including imaginary tangents). 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).

Higher polars

The p-th polar of a C for a natural number p is defined as Δ_Q^pf(x, y, z) = 0. This is a curve of degree n−p. When p is n−1 the p-th polar is a line called the polar line of C with respect to Q. Similarly, when p is n−2 the curve is called the polar conic of C.

Using Taylor series in several variables and exploiting homogeneity, f(λa+μp, λb+μq, λc+μr) can be expanded in two ways as

\mu ^{nf(p,} q, r) + λ \mu ^{n-1} \Delta _{Q} f(p, q, r) + (1)/(2) λ ^{2\mu} ^{n-2} \Delta ^{2} _{Q} f(p, q, r) +... andλ ^{nf(a,} b, c) + \mu λ ^{n-1} \Delta _{P} f(a, b, c) + (1)/(2) \mu ^{2λ} ^{n-2} \Delta ^{2} _{P} f(a, b, c) +... . Comparing coefficients of λ^pμ^n−p shows that(1)/(p!) \Delta ^{p} _{Q} f(p, q, r) = (1)/((n - p)!) \Delta ^{n - p} _{P} f(a, b, c). In particular, the p-th polar of C with respect to Q is the locus of points P so that the (n−p)-th polar of C with respect to P passes through Q.^[1]

Poles

If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1)² poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1. Similarly, the locus of points whose polar lines pass through Q is the first polar of Q and this is also a curve of degree n−1. The polar line of a point is L if and only if it contains both P and Q, so the poles of L are exactly the points of intersection of the two first polars. By Bézout's theorem these curves have (n−1)² points of intersection and these are the poles of L.^[2]

The Hessian

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

\Delta ^{2} _{(x, y, z)} f(a, b, c) = x ^{2{\partial} ^{2} f\over\partial x ^{2}(a,} b, c) +2 xy{\partial ^{2} f\over\partial x\partial y}(a, b, c) +...=0 . The conic is degenerate if and only if the determinant of the Hessian of f,H(f) = \begin{bmatrix} (\partial ^{2} f)/(\partial x ^{2}) & (\partial ^{2} f)/(\partial x\partial y) & (\partial ^{2} f)/(\partial x\partial z) \ \\ (\partial ^{2} f)/(\partial y\partial x) & (\partial ^{2} f)/(\partial y ^{2}) & (\partial ^{2} f)/(\partial y\partial z) \ \\ (\partial ^{2} f)/(\partial z\partial x) & (\partial ^{2} f)/(\partial z\partial y) & (\partial ^{2} f)/(\partial z ^{2}) \end{bmatrix}, vanishes. Therefore, the equation |H(f)|=0 defines a curve, the locus of points whose polar conics are degenerate, of degree 3(n−2) called the Hessian curve of C.

See also

• Polar hypersurface
• Pole and polar

References

• Book: Basset, Alfred Barnard . An Elementary Treatise on Cubic and Quartic Curves. Deighton Bell & Co.. 1901. 16ff..
• Book: Salmon, George. George Salmon

. Higher Plane Curves. Hodges, Foster, and Figgis. 1879. 49ff.. George Salmon.

• Section 1.2 of Fulton, Introduction to intersection theory in algebraic geometry, CBMS, AMS, 1984.

Notes and References

1. Follows Salmon pp. 49-50 but essentially the same argument with different notation is given in Basset pp. 16-17.
2. Basset p. 20, Salmon p. 51