Polar curve

The elliptic curve E : 4Y²Z = X³ − XZ² in blue, and its polar curve (E) : 4Y² = 2.7X² − 2XZ − 0.9Z² 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.

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

${\mathrm{\Delta }}_Q=a\frac{\partial }{\partial x}+b\frac{\partial }{\partial y}+c\frac{\partial }{\partial z}.$

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

$x\frac{\partial f}{\partial x}(p,q,r)+y\frac{\partial f}{\partial y}(p,q,r)+z\frac{\partial f}{\partial z}(p,q,r)=0.$

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

${\mu }^{n}f(p,q,r)+\lambda {\mu }^{n-1}{\mathrm{\Delta }}_{Q}f(p,q,r)+\frac{1}{2}{\lambda }^2{\mu }^{n-2}{\mathrm{\Delta }}_Q^{2}f(p,q,r)+\dots$

and

${\lambda }^{n}f(a,b,c)+\mu {\lambda }^{n-1}{\mathrm{\Delta }}_{P}f(a,b,c)+\frac{1}{2}{\mu }^2{\lambda }^{n-2}{\mathrm{\Delta }}_P^{2}f(a,b,c)+\dots .$

Comparing coefficients of λ^pμ^n−p shows that

$\frac{1}{p!}{\mathrm{\Delta }}_Q^{p}f(p,q,r)=\frac{1}{(n-p)!}{\mathrm{\Delta }}_P^{n-p}f(a,b,c).$

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.

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.

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

${\mathrm{\Delta }}_{(x,y,z)}^{2}f(a,b,c)=x^2\frac{{\partial }^{2}f}{\partial x^2}(a,b,c)+2xy\frac{{\partial }^{2}f}{\partial x\partial y}(a,b,c)+\dots =0.$

The conic is degenerate if and only if the determinant of the Hessian of f,

$H(f)=\left[\begin{array}{ccc}\frac{{\partial }^{2}f}{\partial x^2}& \frac{{\partial }^{2}f}{\partial x\partial y}& \frac{{\partial }^{2}f}{\partial x\partial z}\\ \\ \frac{{\partial }^{2}f}{\partial y\partial x}& \frac{{\partial }^{2}f}{\partial y^2}& \frac{{\partial }^{2}f}{\partial y\partial z}\\ \\ \frac{{\partial }^{2}f}{\partial z\partial x}& \frac{{\partial }^{2}f}{\partial z\partial y}& \frac{{\partial }^{2}f}{\partial z^2}\end{array}\right],$

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

• Basset, Alfred Barnard (1901). An Elementary Treatise on Cubic and Quartic Curves. Deighton Bell & Co.. pp. 16ff.. https://archive.org/details/anelementarytre02bassgoog. 
• {{cite book |title=Higher Plane Curves

|first=George|last=Salmon|publisher=Hodges, Foster, and Figgis|year=1879|pages=49ff. |url=https://archive.org/details/treatiseonhigher00salmuoft

• Section 1.2 of Fulton, Introduction to intersection theory in algebraic geometry, CBMS, AMS, 1984.
• Hazewinkel, Michiel, ed. (2001), "Polar", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=P/p073400 
• Hazewinkel, Michiel, ed. (2001), "Hessian (algebraic curve)", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=H/h047150