- Source: 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
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
Δ
Q
=
a
∂
∂
x
+
b
∂
∂
y
+
c
∂
∂
z
.
{\displaystyle \Delta _{Q}=a{\partial \over \partial x}+b{\partial \over \partial y}+c{\partial \over \partial z}.}
Then ΔQf is a homogeneous polynomial of degree n−1 and ΔQf(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
∂
f
∂
x
(
p
,
q
,
r
)
+
y
∂
f
∂
y
(
p
,
q
,
r
)
+
z
∂
f
∂
z
(
p
,
q
,
r
)
=
0.
{\displaystyle 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 ΔQpf(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
μ
n
f
(
p
,
q
,
r
)
+
λ
μ
n
−
1
Δ
Q
f
(
p
,
q
,
r
)
+
1
2
λ
2
μ
n
−
2
Δ
Q
2
f
(
p
,
q
,
r
)
+
…
{\displaystyle \mu ^{n}f(p,q,r)+\lambda \mu ^{n-1}\Delta _{Q}f(p,q,r)+{\frac {1}{2}}\lambda ^{2}\mu ^{n-2}\Delta _{Q}^{2}f(p,q,r)+\dots }
and
λ
n
f
(
a
,
b
,
c
)
+
μ
λ
n
−
1
Δ
P
f
(
a
,
b
,
c
)
+
1
2
μ
2
λ
n
−
2
Δ
P
2
f
(
a
,
b
,
c
)
+
…
.
{\displaystyle \lambda ^{n}f(a,b,c)+\mu \lambda ^{n-1}\Delta _{P}f(a,b,c)+{\frac {1}{2}}\mu ^{2}\lambda ^{n-2}\Delta _{P}^{2}f(a,b,c)+\dots .}
Comparing coefficients of λpμn−p shows that
1
p
!
Δ
Q
p
f
(
p
,
q
,
r
)
=
1
(
n
−
p
)
!
Δ
P
n
−
p
f
(
a
,
b
,
c
)
.
{\displaystyle {\frac {1}{p!}}\Delta _{Q}^{p}f(p,q,r)={\frac {1}{(n-p)!}}\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
Δ
(
x
,
y
,
z
)
2
f
(
a
,
b
,
c
)
=
x
2
∂
2
f
∂
x
2
(
a
,
b
,
c
)
+
2
x
y
∂
2
f
∂
x
∂
y
(
a
,
b
,
c
)
+
⋯
=
0.
{\displaystyle \Delta _{(x,y,z)}^{2}f(a,b,c)=x^{2}{\partial ^{2}f \over \partial x^{2}}(a,b,c)+2xy{\partial ^{2}f \over \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
)
=
[
∂
2
f
∂
x
2
∂
2
f
∂
x
∂
y
∂
2
f
∂
x
∂
z
∂
2
f
∂
y
∂
x
∂
2
f
∂
y
2
∂
2
f
∂
y
∂
z
∂
2
f
∂
z
∂
x
∂
2
f
∂
z
∂
y
∂
2
f
∂
z
2
]
,
{\displaystyle H(f)={\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.
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.
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) [1994], "Polar", Encyclopedia of Mathematics, EMS Press
Ivanov, A.B. (2001) [1994], "Hessian (algebraic curve)", Encyclopedia of Mathematics, EMS Press
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