Pythagorean-hodograph quintic transition curves of monotone curvature

Abstract

Walton and Meek [Walton, D.J. and Meek, D.S., A Pythagorean-hodograph quintic spiral. Computer Aided Design, 1996, 28, 943–950] have recently advocated the use of Pythagorean-hodograph quintics of monotone curvature, or “PH spirals” for short, as transitional elements that give G 2 connections of linear and circular arcs in applications such as layout of highways and railways—in which context PH curves provide the important advantage of rational offsets and exact rectifications. They construct a PH quintic, interpolating an initial point and tangent and a final tangent, with monotone curvature variation from zero to a given (extremum) final value. We show that using the complex representation for PH curves greatly simplifies this problem and also reveals that the method of Walton and Meek yields a special instance among a one-parameter family of solutions. The additional degree of freedom of PH spirals identified herein relaxes constraints otherwise required to guarantee their existence, and offers the designer precise control over their total length and/or the ability to fine-tune their curvature distributions.