Planar G2 transition with a fair Pythagorean hodograph quintic curve

Abstract

Recently planar cubic and Pythagorean hodograph quintic transition curves that are suitable for G 2 blending were developed. They are suitable for blending, e.g. rounding corners, or for smooth transition between two curves, e.g. two circular arcs. It was shown that a single cubic segment can be used as a transition curve with the guarantee that an S-shaped transition curve will have no curvature extrema, and a C-shaped transition curve will have a single curvature extremum. The results for the cubic curve are now extended to Pythagorean hodograph quintic curves. A Pythagorean hodograph curve has the attractive properties that its arclength is a polynomial of its parameter, and its offset is rational. A quintic is the lowest degree Pythagorean hodograph curve that may have an inflection point. Pythagorean hodograph curves with no curvature extrema for an S-shaped transition, and a single curvature extremum for a C-shaped transition are suitable for the design of fair curves, e.g. in highway design, or for blending in CAD applications.