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Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lee, Sunhong Lee, Hyun Chol Kim, Gwang-Il |
| Copyright Year | 2014 |
| Abstract | and Applied Analysis 3 plane curves with rational offets. These curves are represented in the dual form, in which curves are specified using line coordinates instead of point coordinates. Pottmann showed how to design rational PH curves segments by G1 and G2 Hermite interpolations 17, 18 . However, in our work, what we need is only a suitable PH cubic and a PH-preserving transformation which is algebraically simple as possible and which can generate an extraparameter, and the latter is completely settled by the classical Möbius transformation. Moreover, the transformation is biholomorphic. Thus it preserves the topology of the preimage curve PH cubic . Therefore, the interpolants obtained by our method should have no cusp, although cusps are a generic feature of rational PH curves. They are simple curves or else loops. Hence, to obtain these, even avoiding the easy shortcut, there is no need to follow up the lengthy path with a far starting point. We just use the classical Möbius transformation of PH cubics, that is all. The rest of this paper is organized as follows. In Sections 2 and 3, we review some basic properties of Möbius transformations and planar PH cubics. In Section 4, we solve the C1 Hermite interpolation problem using the Möbius transformations of planar PH cubics. In Section 5, we present the condition on a Hermite dataset, which determine whether the corresponding Hermite interpolant has a loop, we also compare these new interpolants with PH quintics and show that the former have improved stability. We conclude this paper in Section 6. 2. Möbius Transformations A Möbius transformation Φ z is a bijective linear fractional transformation in the extended complex plane C∞ C ∪ {∞}, that is, |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://downloads.hindawi.com/journals/aaa/2012/560246.pdf |
| Alternate Webpage(s) | http://www.maths.tcd.ie/EMIS/journals/HOA/AAA/Volume2012/560246.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
