Interpolation with spatial rational Pythagorean-hodograph curves of class 4.

The paper presents a construction of rational Pythagorean-hodograph curves of class 4 and reveals their properties. In particular, it is shown that each such curve depends on twelve free parameters and has a piecewise rational arc-length function. Geometric interpolation of two data points and two tangent directions is considered in detail and a closed form solution that depends on two shape parameters is given. Regions of shape parameters are derived that imply the interpolant to be regular and admissible. Further, it is shown that one of the shape parameters can be fixed by additionally prescribing also the length of the interpolant. Theoretical results and the construction of G1 Hermite interpolation splines are illustrated with numerical examples.