In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If 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 , given in line coordinates, is known as the tangential equation of . Duality is an involution: the dual of the dual of is the original curve . The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics. Equations Let be the equation of a curve in homogeneous coordinates on the projective plane. Let be the equation of a line, with being designated its line coordinatesin a dual projective plane. The condition that the line is tangent to the curve can be expressed in the form which is the tangential equation of the curve. At a on the curve, the tangent is given by So is a tangent to the curve if Eliminating , , , and from these equations, along with , gives the equation in , and of the dual curve. For example, let be the conic . The dual is found by eliminating , , , and from the equations The first three equations are easily solved for , , , and substituting in the last equation produces Clearing from the denominators, the equation of the dual is Consider a parametrically defined curve in projective coordinates . 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:which in affine coordinates is: 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. From the projective description, one may compute the dual of the dual:which is projectively equivalent to the original curve . Properties of dual curve Properties of the original curve correspond to dual properties on the dual curve. 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. The black