Curve design with more general planar Pythagorean-hodograph quintic spiral segments

•Earlier characterisations of PH quintic spirals are generalised to avoid an endpoint of zero curvature.•The analysis allows the design of C-shaped PH quintic curves without flatspots.•Techniques for designing fair curves using these new spiral segments are presented.

Spiral segments are useful in the design of fair curves. They are important in CAD/CAM applications, the design of highway and railway routes, trajectories of mobile robots and other similar applications. The quintic Pythagorean-hodograph (PH) curve discussed in this article is polynomial; it has the attractive properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. This paper generalises earlier results on planar PH quintic spiral segments and examines techniques for designing fair curves using the new results.