Dual curve

In projective geometry, a dual curve of a given plane curve C 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.

[1] Let f(x, y, z) = 0 be the equation of a curve in homogeneous coordinates on the projective plane.

Let Xx + Yy + Zz = 0 be the equation of a line, with (X, Y, Z) being designated its line coordinates in a dual projective plane.

At a point (p, q, r) on the curve, the tangent is given by So Xx + Yy + Zz = 0 is a tangent to the curve if Eliminating p, q, r, and λ from these equations, along with Xp + Yq + Zr = 0, gives the equation in X, Y and Z of the dual curve.

Its projective tangent line is a linear plane spanned by the point of tangency and the tangent vector, with linear equation coefficients given by the cross product:

In the Introduction image, the red curve has three singularities – a node in the center, and two cusps at the lower right and lower left.

By contrast, on a smooth, convex curve the angle of the tangent line changes monotonically, and the resulting dual curve is also smooth and convex.

If X is a plane algebraic curve, then the degree of the dual is the number of points in the intersection with a line in the dual plane.

Since a line in the dual plane corresponds to a point in the plane, the degree of the dual is the number of tangents to the X that can be drawn through a given point.

If X is smooth (no singular points) then the dual of X has maximum degree d(d − 1).

Geometrically, the map from a conic to its dual is one-to-one (since no line is tangent to two points of a conic, as that requires degree 4), and the 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).

For curves with singular points, these points will also lie on the intersection of the curve and its polar and this reduces the number of possible tangent lines.

The Plücker formulas give the degree of the dual in terms of d and the number and types of singular points of X.

The dual can be visualized as a locus in the plane in the form of the polar reciprocal.

This is defined with reference to a fixed conic Q as the locus of the poles of the tangent lines of the curve C.[2] The conic Q is nearly always taken to be a circle, so the polar reciprocal is the inverse of the pedal of C. Similarly, generalizing to higher dimensions, given a hypersurface, the tangent space at each point gives a family of hyperplanes, and thus defines a dual hypersurface in the dual space.

For any closed subvariety X in a projective space, the set of all hyperplanes tangent to some point of X is a closed subvariety of the dual of the projective space, called the dual variety of X.

Examples 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.

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.