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. If C is algebraic then so is its dual and the degree of the dual is known as the class of the original curve. The equation of the dual of C, given in line coordinates, is known as the tangential equation of C.

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

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.

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