Dual Curve - Equations

Equations

Let f(x, y, z)=0 be the equation of a curve in homogeneous coordinates. Let Xx+Yy+Zz=0 be the equation of a line, with (X, Y, Z) being designated its line coordinates. The condition that the line is tangent to the curve can be expressed in the form F(X, Y, Z)=0 which is the tangential equation of the curve.

Let (p, q, r) be the point on the curve, then the equation of the tangent at this point 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.

For example, let C be the conic ax2+by2+cz2=0. Then dual is found by eliminating p, q, r, and λ from the equations

The first three equations are easily solved for p, q, r, and substituting in the last equation produces

Clearing 2λ from the denominators, the equation of the dual is

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.