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

Authors: Zbynek Sír and Bert Jüttler
Journal: Math. Comp. 76 (2007), 1373-1391
MSC (2000): Primary 68U07; Secondary 53A04, 65D17
Published electronically: February 1, 2007
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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

Zbynek Sír
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

Bert Jüttler
Affiliation: Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, 4040 Linz, Austria

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.

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