First order hermite interpolation with spherical Pythagorean-hodograph curves

Abstract

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC1 Hermite data, we construct a spatial PH curve on a sphere that is aC1 Hermite interpolant of the given data as follows: First, we solveC1 Hermite interpolation problem for the stereographically projected planar data of the given data in ℝ3 with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ℝ3 using the inverse general stereographic projection.