Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves
type: Item
instance of: scholarly article
title: Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves
zbMATH Open document ID: 1001.41003
DOI: 10.1023/A:1016280811626
author: Rida T. Farouki
author: Mohammad al-Kandari
author: Takis Sakkalis
published in: Advances in Computational Mathematics
publication date: 2002-08-19T00:00:00Z
review text: A parametric curve \(\mathbf{r}(t)=(x_1(t),x_2(t),x_3(t))\) is a Pythagorean-hodograph (PH) space curve if each component \(x_i\) is a polynomial and the derivative \(\mathbf{r}^\prime\) satisfies the Pythagorean condition \(\sum (x_i^\prime(t))^2=\sigma^2(t)\). The purpose of this paper is to solve the Hermite interpolation problem using quaternion representation for spatial PH curves with hodographs of the form \(x^\prime(t)=u^2(t)+v^2(t)-p^2(t)-q^2(t)\), \(y^\prime(t)=2[u(t)q(t)+v(t)p(t)]\), \(z^\prime(t)=2[v(t)q(t)-u(t)p(t)]\), \(\sigma(t)=u^2(t)+v^2(t)+p^2(t)+q^2(t)\). The complex-variable model is here considered to facilitate the solution procedure and analysis of the resulting interpolants. It is formulated the first-order Hermite interpolation for PH quintic space curves in scalar form, and the corresponding quaternion version to solve the problem of characterizing the set of spatial rotations by certain combinations of four polynomials which admit a characterization in terms of quaternions. This solution is applied to the construction of spatial PH quintic Hermite interpolants. The quaternion solution of the first order Hermite interpolation problem considered in this paper, admits a natural extension to the construction of spatial \(C^2\) PH quintic splines which interpolate a sequence of points in space with prescribed end conditions. Examples of rotation-invariant PH quintic space curves are given.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 1783438
zbMATH Keywords: Pythagorean-hodograph curves
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: quaternions
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: a:1016280811626
OpenAlex ID: W2346131912
description: scientific article; zbMATH DE number 1783438