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. Duality is an involution: the dual of the dual of C is the original curve C.
The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics.[1]
- ^ See (Arnold 1988)