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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

$C^2$ Hermite interpolation by Pythagorean Hodograph space curves


Authors: Zbyněk Šír and Bert Jüttler
Journal: Math. Comp. 76 (2007), 1373-1391
MSC (2000): Primary 68U07; Secondary 53A04, 65D17
DOI: https://doi.org/10.1090/S0025-5718-07-01925-4
Published electronically: February 1, 2007
MathSciNet review: 2299779
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Abstract: We solve the problem of $C^2$ Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of $C^2$ space boundary data (two points with associated first and second derivatives) we construct a four–dimensional family of PH interpolants of degree $9$ and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible approximation order 6. Third, it is symmetric in the sense that the interpolant of the “reversed” set of boundary data is simply the “reversed” original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree $9$ which is globally $C^2$, and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and approximation order.


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Additional Information

Zbyněk Šír
Affiliation: Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, 4040 Linz, Austria
Address at time of publication: Charles University, Sokolovská 83, 18675 Prague, Czech Republic
Email: zbynek.sir@jku.at

Bert Jüttler
Affiliation: Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, 4040 Linz, Austria
Email: bert.juettler@jku.at

Keywords: Pythagorean Hodograph curves, Hermite interpolation, G-code, approximation order
Received by editor(s): May 24, 2005
Received by editor(s) in revised form: October 27, 2005
Published electronically: February 1, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.