Shape-preserving interpolation of spatial data by Pythagorean-hodograph quintic spline curves

Abstract

The interpolation of discrete spatial data-a sequence of points and unit tangents-by G(1) Pythagorean-hodograph (PH) quintic spline curves, under shape constraints, is addressed. To achieve this, a local Hermite scheme incorporating a tension parameter for each spline segment is employed, the imposed shape constraints being concerned with preservation of convexity at the knots and the sign of the discrete torsion over each spline segment. An asymptotic analysis in terms of the tension parameters is developed, and it is shown that satisfaction of the prescribed shape constraints can always be achieved for each spline segment by a suitable choice of the free angular parameters that characterize each PH quintic Hermite segment. In particular, it is proved that the cubic-cubic criterion (Farouki, R. T., Giannelli, C., Manni, C. & Sestini, A. (2008) Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures. Comput. Aided Geom. Design, 25, 274-297; Sestini, A., Landolfi, L. & Manni, C. (2013) On the approximation order of a space data dependent PH quintic Hermite interpolation scheme. Comput. Aided Geom. Design, 30, 148-158) for specifying these free parameters ensures satisfaction of the desired shape-preserving properties, requiring only mild application of the tension parameters that does not compromise the overall fairness of the interpolant. The performance of the method is illustrated through some computed examples.