Synthesis of plane curves with prescribed local geometric properties using periodic splines

Abstract

A wide class of tecnical problems involves the determination of shape complying with certain local geometric conditions, such as prescribed — or bounded — values of slope and curvature. Traditionally these problems have been solved by resorting to well-known curves described by algebraic formulae. The class of such curves available is limited, however, and in general curves comply with the desired conditions only at certain intervals. By the introduction of periodic splines it is shown how curves can be synthesized that meet local geometric conditions at a finite set of points that may be located arbitrarily throughout the whole interval of interest. The problem is then reduced to one of solving a system of algebraic equations that can be either linear or nonlinear. The method proposed here is illustrated with problems arising from several engineering disciplines.