Paper Info
Title
Singular Cases Of Planar And Spatial C-1 Hermite Interpolation Problems Based On Quintic Pythagorean-Hodograph Curves
Abstract
A well-known feature of the Pythagorean-hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first-order Hermite data, and a two-parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a "double" solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics. (C) 2020 Elsevier B.V. All rights reserved.
Year
DOI
Venue
2020
10.1016/j.cagd.2020.101930
COMPUTER AIDED GEOMETRIC DESIGN
Keywords
DocType
Volume
Pythagorean-hodograph curves, Algebraic equations, Hermite interpolation, Complex numbers, Quaternions, Arc length
Journal
82
ISSN
Citations 
PageRank 
0167-8396
0
0.34
References 
Authors
0
3
Name
Order
Citations
PageRank
R. T. Farouki1575125.58
Kai Hormann2184.35
Federico Nudo300.34