Dual curve

Curves, dual to each other

In projective geometry, a dual curve of a given plane curve is a duality partner of a given curve. The dual of a projective plane is naturally identified with the space of lines in the projective plane. Thus if C is a curve in a projective plane, then the dual curve is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line.

The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics.

Parametric equations

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.

Degree

Smooth curves

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).

If d > 2, then d−1 > 1 so d(d − 1) > d, and thus the dual curve must be singular, by duality, otherwise the bidual would have higher degree than the original curve.

For a smooth curve of degree d = 2, then the degree of the dual is also 2: the dual of a conic is a conic. This can also be seen geometrically: the map from a conic to its dual is 1-to-1 (since no line is tangent to two points of a conic, as that requires degree 4), and tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires degree 3).

Finally, the dual of a line (a curve of degree 1) is a point (the tangent line is the same at all points, and agrees with the line itself), whose dual is the original line.

Singular curves

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.^{[citation needed]}

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'.

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 the distance from 0 to p, and a' is a distance from 0 to p', then a' = 1 / a, p' = p / a² 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.

Generalizations

Higher dimensions

Similarly, generalizing to higher dimensions, given a hypersurface, the tangent plane at each point gives a family of hyperplanes, and thus defines a dual hypersurface in the dual space.

Dual polygon

The dual curve construction works even if the curve is piecewise linear (or piecewise differentiable, but the resulting map is degenerate (if there are linear components) or ill-defined (if there are singular points).

In the case of a polygon, all points on each edge share the same tangent line, and thus map to the same vertex of the dual, while the tangent line of a vertex is ill-defined, and can be interpreted as all the lines passing through it, with angle between the two edges – regardless, the map from the vertex is ill-defined. This accords both with projective duality (lines map to points, and points to lines), and with the limit of smooth curves with no linear component: as a curve flattens to an edge, its tangent lines map to closer and closer points; as a curve sharpens to a vertex, its tangent lines spread further apart.

See also

• Dual polygon
• Plücker formula
• Hough transform
• Gauss map

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