Dual Curve
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 coordinates in 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 ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   [Amazon]