Dual curve

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

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

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 R², 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 x² + y² = 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'. (Picture needed!!)

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, as vectors, and, if p = (p₁,p₂), then l has an equation p₁x + p₂y = a²,
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 R² and its dual.

[edit] See also

• Plücker formula
• Hough transform
• Gauss map

[edit] References

• Fulton. Intersection Theory. 
• Walker. Algebraic Curves. 
• Brieskorn, Knorrer. Plane algebraic curves. 

Differential transforms of plane curves

Parallel curve | Evolute | Involute | Pedal curve | Contrapedal curve | Negative pedal curve | Dual curve

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