Contact Transformation

a transformation of plane curves such that two mutually tangent curves are transformed into two other curves that are also tangent to each other. A contact transformation is defined by the formulas

(*) X = f (x, y, yʹ) Y = ɸ(x, y, yʹ)

where x and y are the coordinates of the variable point of a curve and X and Y are the coordinates of the variable points of the curve’s image. In order that the formula (*) define a contact transformation, Yʹ = dY/dX must be independent of y” = d²y/dx². Examples of contact transformations are point transformations, which are defined by the formulas

X = f (x, y) Y = ɸ(x, y)

and the Legendre transformations.

Contact transformations are used in the theory of differential equations and in differential geometry. The general theory of contact transformations was developed by M. S. Lie. Contact transformations of surfaces in space are defined in an analogous fashion.