Planar C1 Hermite interpolation with uniform and non-uniform TC-biarcs

Pythagorean hodograph curves (shortly PH curves), introduced in Farouki and Sakkalis (1990), form an important subclass of polynomial parametric curves and currently represent standard objects in geometric modelling. In this paper, we focus on Tschirnhausen cubic as the only one Pythagorean hodograph cubic and we study planar C^1 Hermite interpolation with two arcs of Tschirnhausen cubic joined with C^1 continuity (the so-called TC-biarc). We extend results presented in Farouki and Peters (1996) in several ways. We study an asymptotical behaviour of the conversion of an arbitrary planar curve with well defined tangent vectors everywhere to a C^1 PH cubic spline curve and we prove that the approximation order is 3. Further, we analyze the shape of TC-biarcs and provide a sufficient condition for input data guaranteeing TC-biarc without local and pairwise self-intersections. Finally, we generalize the basic uniform method to the non-uniform case, which introduces a free shape parameter, and we formulate an algorithm for a suitable choice of this shape parameter such that the corresponding non-uniform TC-biarc is without local and pairwise self-intersections (if such a parameter exists).