Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics
We present an algorithm for C1 Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solve C1 Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/560246 |
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| author | Sunhong Lee Hyun Chol Lee Mi Ran Lee Seungpil Jeong Gwang-Il Kim |
| author_facet | Sunhong Lee Hyun Chol Lee Mi Ran Lee Seungpil Jeong Gwang-Il Kim |
| author_sort | Sunhong Lee |
| collection | DOAJ |
| description | We present an algorithm for C1 Hermite interpolation
using Möbius transformations of planar polynomial Pythagoreanhodograph
(PH) cubics. In general, with PH cubics, we cannot
solve C1 Hermite interpolation problems, since their lack of parameters
makes the problems overdetermined. In this paper, we
show that, for each Möbius transformation, we can introduce an
extra parameter determined by the transformation, with which we
can reduce them to the problems determining PH cubics in the
complex plane ℂ. Möbius transformations preserve the PH property
of PH curves and are biholomorphic. Thus the interpolants
obtained by this algorithm are also PH and preserve the topology
of PH cubics. We present a condition to be met by a Hermite
dataset, in order for the corresponding interpolant to be simple or
to be a loop. We demonstrate the improved stability of these new
interpolants compared with PH quintics. |
| format | Article |
| id | doaj-art-78033096f8d444d9a9823463936a88f8 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-78033096f8d444d9a9823463936a88f82025-02-03T01:30:29ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/560246560246Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph CubicsSunhong Lee0Hyun Chol Lee1Mi Ran Lee2Seungpil Jeong3Gwang-Il Kim4Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of KoreaDepartment of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of KoreaDepartment of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of KoreaDepartment of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of KoreaDepartment of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of KoreaWe present an algorithm for C1 Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solve C1 Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Möbius transformation, we can introduce an extra parameter determined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex plane ℂ. Möbius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.http://dx.doi.org/10.1155/2012/560246 |
| spellingShingle | Sunhong Lee Hyun Chol Lee Mi Ran Lee Seungpil Jeong Gwang-Il Kim Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics Abstract and Applied Analysis |
| title | Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics |
| title_full | Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics |
| title_fullStr | Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics |
| title_full_unstemmed | Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics |
| title_short | Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics |
| title_sort | hermite interpolation using mobius transformations of planar pythagorean hodograph cubics |
| url | http://dx.doi.org/10.1155/2012/560246 |
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