# Polar curve

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

LetCbe defined in homogeneous coordinates byf(x, y, z) = 0 wherefis a homogeneous polynomial of degreen, and let the homogeneous coordinates ofQbe (a,b,c). Define the operator

Then Δ_{Q}*f* is a homogeneous polynomial of degree *n*−1 and Δ_{Q}*f*(*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

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*. Note also that 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}^{p}*f*(*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*(λ*a*+μ*p*, λ*b*+μ*q*, λ*c*+μ*r*) can be expanded in two ways as

and

Comparing coefficients of λ^{p}μ^{n−p} shows that

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*, the *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*. So 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

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

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

## References

- Basset, Alfred Barnard (1901).
*An Elementary Treatise on Cubic and Quartic Curves*. Deighton Bell & Co. pp. 16ff. - Salmon, George (1879).
*Higher Plane Curves*. Hodges, Foster, and Figgis. pp. 49ff. - Section 1.2 of Fulton,
*Introduction to intersection theory in algebraic geometry*, CBMS, AMS, 1984. - Ivanov, A.B. (2001), "Polar", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Ivanov, A.B. (2001), "Hessian (algebraic curve)", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4