The conformal map z → z2 of the hodograph plane

Abstract

Representing plane parametric curves by complex-valued functions of a real parameter, a simple conformal map of the hodograph plane serves to elucidate the relationship between Pythagorean-hodograph (PH) curves and polynomial curves in general (PH curves arc a special class of polynomial curves, distinguished by the property that their are lengths and offset curves can be computed rationally). For the case of regular parameterizations, a one-to-one correspondence between the PH curves and general polynomial curves is established, and relationships concerning the tangents, curvatures, inflections, arc lenghts, and other intrinsic properties of corresponding curves are identified. The use of complex representations also greatly facilitates the derivation of constraints on the Bézier control polygons of regular polynomial curves that are sufficient and necessary for a Pythagorean hodograph.