Planar G2 transition with a fair Pythagorean hodograph quintic curve

Recently planar cubic and Pythagorean hodograph quintic transition curves that are suitable for G2 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.