Nonparametric Bézier Representation of Polynomial Transition Curves

Abstract

In a series of papers in this journal, polynomial solutions of transition curves are introduced for road design, constructed as graphs of polynomial functions. These curves admit a simple expression in nonparametric Bézier form, which facilitates their implementation into commercial software and data exchange. Furthermore, in this Bézier representation, few compact formulas describe all possible cases, and the boundary conditions defining the curves translate into intuitive geometric arrangements of the control points. This is illustrated with several examples, including cases where, at the endpoints, zero curvature and smoothness of the curvature diagram are required. The authors also consider an alternative using a nonparametric B-spline curve of lower degree, but at the cost of a more complex model. Quartic splines, at a minimum, are required to ensure a continuous change of lateral acceleration along the transition.