A fully data--dependent criterion for free angle selection in spatial PH cubic biarc Hermite interpolation

Cubic biarcs are the natural counterpart, in the context of the PH spatial Hermite interpolation, of piecewise quadratics for Hermite interpolation of standard parametric curves. Recently, it has been shown that, spatial Hermite interpolation can be always treated with PH cubic biarcs, providing an interesting alternative to the use of higher degrees, see Bastl et al.(2014). Beside the parameter value at the joint of the two arcs, which is typically free in any scheme of this kind, two free angular parameters remain. They should be automatically fixed by some suitable criterion in order to ensure the generation of interpolants with a reasonable shape. In this paper, an alternative to the constant choice of free parameters presented in Bastl et al.(2014)is proposed. It consists of an extension of the so called CC selection strategy introduced in Farouki et al.(2008)to the biarc setting. Such strategy is fully data-dependent, does not require any special configuration of the coordinate system and it guarantees the PH cubic reproduction when such kind of the interpolant exists. Moreover, the obtained scheme possesses other two important features, i.e. it is third order accurate and gives the possibility to control the torsion sign of the produced interpolant, in case with the introduction of an additional tension parameter relaxing the smoothness from C1 to G1. The numerical results, based also on the straightforward spline extension of the scheme, confirm the developed theoretical analysis.