Dual curve

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Curves, dual to each other

In projective geometry, a dual curve of a given curve is a duality partner of a given curve. The curve is formed by taking the set of lines tangent to the curve. Then the dual of each tangent line will give a point, and the set of all points will form the dual curve.

For a parametrically defined curve its dual curve is defined by the following parametric equations:

$X[x,y]=\frac{y'}{yx'-xy'}$

$Y[x,y]=\frac{x'}{xy'-yx'}$

The dual of an inflection point will give a cusp and two points sharing the same tangent line will give a self intersection point on the dual.

If X is a smooth plane algebraic curve of degree d, then the dual of X is a (usually singular) plane curve of degree $d (d - 1)$ (cf. Fulton, Ex. 3.2.21).

For an arbitrary plane algebraic curve X of degree d, its dual is a plane curve of degree $d (d - 1) - δ$ , where $δ$ is the number of singularities of X counted with certain multiplicities: each node is counted with multiplicity 2 and each cusp with multiplicity 3 (cf. Fulton, Ex. 4.4.4).

It is known that the bi-dual curve to an algebraic curve X is isomorphic to X in characteristics 0.

[edit] Classical construction

If C is a curve in a real euclidian plane $R 2$ , there is a beautiful classical construction of a dual curve C'. (cf., for example, [Brieskorn, Knorrer]). It uses the notion of inversion and polar curve.

Let S be a (real) unit circle $x 2 + y 2 = 1$ , and assume we are given a point p inside S. Let l be a line through p orthogonal to the radius through p. The line l intersects S at two points, say, a and b. Let p' be the intersection point of tangents to S at the points a and b. Then p and p' are said to be inverse to each other with respect to the circle S. Let l' be a line through p' parallel to l. Then the line l' is said to be polar to the circle S with respect to the point p, and l is said to be polar to S with respect to the point p'.

Now, if p is on a curve C, and l is tangent to C at p, then one can see that p' is a point of the dual curve C'. The converse is also true: if l' is tangent to C at p', then p is a point of C'.

Algebraically, if a is a distance from 0 to p, and a' is a distance from 0 to p', then $a' = 1 / a$ , $p' = p / a 2$ as vectors, and, if $p = (p 1, p 2)$ , then $l$ has an equation $p 1 x + p 2 y = a 2,$
From the last formula one can see that $p = [l]$ , i.e., p is the class of the line l if we identify the plane $R 2$ and its dual.

[edit] See also

[edit] References

Fulton, William (1998), Intersection Theory, Springer-Verlag, ISBN 978-3-540-62046-4
Walker, R.J. (1950), Algebraic Curves, Princeton
Brieskorn, E. & Knorrer, H. (1986), Plane Algebraic Curves, Birkhauser, ISBN 978-3-764-31769-0