A new method in highway route design: joining circular arcs by a single C-Bézier curve with shape parameter

Abstract

We constructed a single C-Bezier curve with a shape parameter for G2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the Bezier curves, has no curvature extrema, while a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms were presented in detail, strict mathematical proofs were given, and the effectiveness of the method was shown by examples. This method has the following three advantages: (1) the pattern is unified; (2) the parameter able to adjust the shape of the transition curve is available; (3) the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. These advantages make the method simple and easy to implement.