Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

Abstract

It is shown that, depending upon the orientation of the end tangents t 0 , t 1 relative to the end point displacement vector p = p 1 − p 0 , the problem of G 1 Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p 0 , . . . , p n in R 3 , compatible with a G 1 piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.

An approach to geometric interpolation by Pythagorean-hodograph curves

“…As a result, geometrically intuitive necessary and sufficient conditions for the existence of solutions are obtained. Although this interpolation problem has already been considered in [16,22,23], the results obtained there do not cover all the data configurations where the solutions exist. However, recently in [20], a complete characterization has been provided and conditions for the existence of the solution were independently derived for all possible data.…”

A characterization of helical polynomial curves of any degree

“…In [8] the following Hermite interpolation problem is considered: Given end points p 0 and p 1 and end tangents − → t 0 and − → t 1 satisfying (…”

“…According to this approach, the already known results ( [7,8]) about Hermite interpolation using helical curves are revisited and their proofs simplified. Moreover, the classification into two classes of helical PH quintics introduced in [6] is also revisited, and a result concerning a sufficient condition on the quaternions generating a PH helical curve is introduced (see Cor.…”

Pythagorean-Hodograph Cubics and Geometric Hermite Interpolation

“…To reach the goal, we invoke the characterization of the planar PH curves and Theorem about the planar PH cubics, given by Farouki and Sakkalis ( [6]). For spatial PH cubics, Pelosi et al [10] partially gave the geometric Hermite interpolants. For Minkowski PH cubics, Kosinka and Jütter [7] solved the geometric Hermite interpolation problem.…”