Deformation of plane curves preserving Pythagorean hodographs

Parametric plane curves are expressed as complex valued functions of a real parameter. The method is convenient for investigating Pythagorean-hodograph curves. A complex function transforms one curve to another curve in the complex plane. There are complex functions which preserve the Pythagorean hodographs of the curves. These functions can be used in deforming Pythagorean-hodograph curves. In particular, the Mobius transformation is useful in controlling the deformation.