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

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- by Zbyněk Šír and Bert Jüttler PDF
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**76**(2007), 1373-1391 Request permission

## 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.## References

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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
- Received by editor(s): May 24, 2005
- Received by editor(s) in revised form: October 27, 2005
- Published electronically: February 1, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2299779