Polar curve#The Hessian

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

:$\backslash Delta\_Q\; =\; a\{\backslash partial\backslash over\backslash partial\; x\}+b\{\backslash partial\backslash over\backslash partial\; y\}+c\{\backslash partial\backslash over\backslash partial\; z\}.$

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

:$x\{\backslash partial\; f\backslash over\backslash partial\; x\}(p,\; q,\; r)+y\{\backslash partial\; f\backslash over\backslash partial\; y\}(p,\; q,\; r)+z\{\backslash partial\; f\backslash over\backslash partial\; z\}(p,\; q,\; r)=0.$

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.

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).

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

:$\backslash mu^nf(p,\; q,\; r)\; +\; \backslash lambda\backslash mu^\{n-1\}\backslash Delta\_Q\; f(p,\; q,\; r)\; +\; \backslash frac\{1\}\{2\}\backslash lambda^2\backslash mu^\{n-2\}\backslash Delta\_Q^2\; f(p,\; q,\; r)+\backslash dots$

and

:$\backslash lambda^nf(a,\; b,\; c)\; +\; \backslash mu\backslash lambda^\{n-1\}\backslash Delta\_P\; f(a,\; b,\; c)\; +\; \backslash frac\{1\}\{2\}\backslash mu^2\backslash lambda^\{n-2\}\backslash Delta\_P^2\; f(a,\; b,\; c)+\backslash dots\; .$

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

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

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.Follows Salmon pp. 49-50 but essentially the same argument with different notation is given in Basset pp. 16-17.

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.Basset p. 20, Salmon p. 51

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

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

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

:$H(f)\; =\; \backslash begin\{bmatrix\}$

\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{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.

• Polar hypersurface
• Pole and polar

{{reflist}}

• {{cite book |title=An Elementary Treatise on Cubic and Quartic Curves

|first=Alfred Barnard |last=Basset|publisher=Deighton Bell & Co.|year=1901|pages=16ff.

|url=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

|authorlink=George Salmon}}

• Section 1.2 of Fulton, Introduction to intersection theory in algebraic geometry, CBMS, AMS, 1984.
• {{springer|title=Polar|id=P/p073400|last=Ivanov|first=A.B.}}
• {{springer|title=Hessian (algebraic curve)|id=H/h047150|last=Ivanov|first=A.B.}}

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