Pythagorean hodograph spline spirals that match G3 Hermite data from circles

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.

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