Pythagorean hodograph spline spirals that match $G^3$ Hermite data from circles

Abstract

A construction is given for a $G^3$ piecewise rational Pythagorean hodograph convex spiral which interpolates two $G^3$ Hermite data associated with two non-concentric circles, one being inside the other. The spiral solution is of degree 7 and is the involute of a $G^2$ convex curve, referred to as the evolute solution, with prescribed length, and composed of two PH quartic curves. Conditions for $G^3$ continuous contact with circles are then studied and it turns out that an ordinary cusp at each end of the evolute solution is required. Thus, geometric properties of a family of PH polynomial quartics, allowing to generate such an ordinary cusp at one end, are studied. Finally, a constructive algorithm is described with illustrative examples.