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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 560246, 15 pages
Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics
Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea
Received 17 January 2012; Accepted 11 February 2012
Academic Editor: Saminathan Ponnusamy
Copyright © 2012 Sunhong Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present an algorithm for Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solve 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 . 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.
Farouki and Sakkalis  introduced Pythagorean-hodograph (PH) curves, which are a special class of polynomial curves with a polynomial speed function. These curves have many computationally attractive features: in particular, their arc lengths and offset curves can be determined exactly. Farouki  reviews the abundant results on these curves obtained by many researchers. Hermite interpolation with PH curves is one of the main topics in this research (Farouki and Neff , Albrecht and Farouki , Jüttler , Jüttler and Mäurer , Farouki et al. , Pelosi et al. , and Šír et al. ).
In this paper, we solve the Hermite interpolation problem using the Möbius transformations of polynomial PH cubics in the plane. The use of Möbius transformation has been demonstrated in recent publications [10, 11]. In , Bartoň et al. used a general Möbius transformation in , which is defined as a composition of an arbitrary number of inversions with respect to spheres or planes. They showed that is a rational PH curve for any general Möbius transformation , if is a polynomial PH space curve in . (The preservation of PH properties under transformation is first studied by Ueda .) They also presented an algorithm for Hermite interpolation. In this work, we use the classical Möbius transformation, a bijective linear fractional transformation in the extended complex plane , that is, for some complex numbers , , , and for which . Using this transformation, we can solve the Hermite interpolation problems with PH cubics. In general, with PH cubics, we cannot solve Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. But we can show that, for a Hermite interpolation problem, we are always able to obtain four interplants which are constructed by PH cubics. The Möbius transformation makes this possible, since it permits the introduction of a new extra parameter into the problem, which is to be reduced to a simple problem to determine PH cubics as follows: here we adapt the complex representation method  to solve the Hermite interpolation problem. The original problem is, for a Hermite dataset , to find a polynomial PH curve and a Möbius transformation , which satisfy Next, by an appropriate translation, rotation, and scaling of the dataset, we can arrange that and and take a Möbius transformation which fixes 0 and 1, for some nonzero complex number . Then the inverse image of the Hermite dataset under a Möbius transformation makes (1.2) into which are suitable forms for adapting the complex representation method (for details, see Section 4). Farouki and Neff  already solved the Hermite interpolation problem with PH quintics. According to (1.2), this is exactly the case in which is a quintic and is the identity, that is, . On the other hand, our work in this paper is the case just when is not the identity, that is, . At the end of this paper, we will compare our interpolants with PH quintic ones for the same Hermite dataset.
The interpolants obtained by our method are specific rational curves represented by complex rational functions. For planar rational curves, there already exists a general theory, which were introduced by Pottmann  and Fiorot and Gensane : they studied rational 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 and 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 extra-parameter, 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 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 is a bijective linear fractional transformation in the extended complex plane , that is, for some complex numbers , , , and for which . Then is a one-to-one correspondence on the extended complex plane with its inverse The derivative of is For any Möbius transformations and , is also a Möbius transformation. Thus the set of all Möbius transformations forms a group under composition.
A rational plane curve is called a Pythagorean-hodograph (PH) curve  if there exists a rational function such that
Lemma 2.1. Let be a Möbius transformation and be a polynomial PH curve. Then is a rational PH curve.
Proof. Since , we have This completes the proof.
For a polynomial PH curve of degree , a Möbius transformation of is a rational curve, also of degree , with coefficients in the complex plane . However, if we associate the complex plane with , and express as a rational curve with real coefficients in then the result is generally a rational PH curve of degree or . If we perform a further Möbius transformation , the rational curve retains a degree of or , since is a Möbius transformation.
Lemma 2.2. Let be a Möbius transformation of a polynomial curve , such that and . Then there exist a polynomial curve and a Möbius transformation , such that , , and .
Proof. We can find a Möbius transformation for some complex constants and such that and is a polynomial curve with and . Consequently, we can obtain the Möbius transformation such that .
A Möbius transformation of this sort also fixes 0 and 1.
Lemma 2.3. Let be a Möbius transformation with and . Then there exists a nonzero complex constant such that If and , then , where
Proof. Let be a Möbius transformation with and . Then from we get , and from we get . Thus , where . Let and . Then we obtain , where and .
Now let be a Möbius transformation with and . Let , so that and . Then, since and , we get , where for some nonzero . Thus we obtain Moreover, in the same way, we can obtain and , so that .
Since , we have and .
3. Planar Pythagorean-Hodograph Cubics
A planar polynomial curve is a PH curve  if and only if there exist polynomials , , and , which satisfy Note that, if , then . In this paper, we will assume that is monic, meaning that its leading coefficient is 1.
A polynomial curve is a PH curve  if and only if there exists a polynomial and a polynomial curve such that Suppose that the PH cubic is a line. Then the hodograph can be expressed as , where is a nonzero point and is the quadratic monic polynomial and , , and are real constants such that .
Let be a PH cubic for which . Since is linear, we can write in Bernstein form: where and are distinct complex constants. The hodograph can then be expressed as If we represent the PH cubic in the Bernstein form then we obtain where can be chosen arbitrarily.
4. First-Order Hermite Interpolation
We will now solve the Hermite interpolation problem using Möbius transformations of PH cubics.
Let and be the initial and final points to be interpolated, where . Let and , respectively, be the initial vector at and the final vector at , where and . For this Hermite dateset , we want to find planar PH cubics and Möbius transformations which satisfy (1.2), which are equivalent to By an appropriate translation, rotation, and scaling of the data-set, we can arrange that and . Then, from Lemmas 2.2 and 2.3, we seek some nonzero constants and PH cubics , which satisfy (1.4).
4.1. Case of
In this case, (4.1) become From the second and third of these equations, we can see that Hermite interpolants exist if and only if for some integers . In this case, for or , where is any positive number, we have Consequently, we can obtain the PH cubics and their Möbius transformations
4.2. Case of Where
From (3.5) and (3.6), (4.1) become If we let and also let , and , then second and third equations in (4.6) imply that or , and so we have Now let where if , and if . Then we have and , or and . Consequently, we can obtain the four PH cubics where or . Note that if and only if is −1 or . From the PH cubics we can obtain the Möbius transformations of the PH cubics , where If is nonreal, then both and are nonlinear. But if is a real number, then if and only if both and are linear.
We can summarize these results.
Theorem 4.1. Let be a Hermite data-set such that and . (a)Let be the vector given by (4.7), and let be or . Then all Hermite interpolants using Möbius transformations of planar PH cubics , such that for some linear curve , are , from (4.10) and (4.11), where and are given by (4.9).(b) Hermite interpolants using Möbius transformations of planar PH cubics , such that for some real number , and such that , exist if and only if for some integers . In this case, the interpolants are given by (4.5), where is given by (4.3) and (4.4), where or for any positive number .
5. Best Interpolant
In this section we consider how to choose the best interpolant for a given Hermite data-set .
We will begin by presenting a condition under which the Möbius transformation of a PH cubic has a loop, where for some distinct complex constants and . Since represents a one-to-one correspondence on the extended complex plane, the condition that has a loop is both necessary and sufficient to establish that has a loop. Under the conditions and , the PH cubic is given by , where The condition that there exist constants and , such that and , is necessary and sufficient to establish that has a loop. From , we can obtain , which implies This equation is equivalent to and hence Consequently, has a loop if and only if , (see Figure 1) where On the other hand, the PH cubic can be represented by where , , and . From and , we can obtain and hence Note that Therefore we conclude as following.
Theorem 5.1. Suppose that is a Möbius transformation of a planar PH cubic, such that , and for some distinct complex constants and (see Figure 2). Then is a simple curve if and only if , where
For a given Hermite data-set , the term in (4.7) belongs to if and only if and have a loop; and belongs to if and only if and have a loop. Note that is a subset of the left half-plane, that is, . Thus we can deduce that both and , or both and are simple curves. From these simple curves we can choose a best interpolant, which is that with the least bending energy where is the curvature of .
Example 5.2. Consider a Hermite data-set . Then the vector becomes Thus and , which implies that and are simple but and each have a loop. See Figure 3.
Example 5.3. In the case of a Hermite data-set , the vector becomes Thus and , which implies that , , , and are all simple. See Figure 4.
Example 5.4. Consider a family ofHermite data-sets, where. We construct Hermite interpolants that satisfy these data-sets using Möbius transformations of PH cubics, and also PH quintics, all shown in Figure 5. The Möbius transformations of the PH cubics always provide two -shaped simple curves and two other curves; the latter are -shaped simple curves whenor and have a single loop in the other cases. As the parametric speed of the initial Hermite condition increases, the -shaped interpolants change from simple curves to single loops, while the simple -shaped interpolants retain their original shape characteristics. We also observe that, unlike the -shaped interpolants produced by Möbius transformations of PH cubics, the -shaped PH quintic interpolants may be simple (like the curve labeled 4 in Figure 5), or have one or two loops (some PH quintics labeled 2 in Figure 5 are -shaped double loops).
We observed the behavior of these interpolants as the parametric speed at the endpoints changes. As this speed increases, the arc-lengths of PH quintics increase rapidly, but the arc-length, of Möbius transformations of PH cubics are generally less affected. In particular, the simple -shaped interpolants, produced by Möbius transformations of PH cubics show little change in arc-length. Table 1 shows that these latter interpolants have both lower bending energies and shorter arc-lengths, than all the other interpolants we are considering. If we look at Table 1 and identify the most shapely interpolants with the lowest bending energies, we find that the best Möbius transformation of a PH cubic is always -shaped and simple. However, the merit of the PH quintic interpolants depends on the parametric speeds at their end points. For example, in Figures 5(a)′ and 5(b)′, the curves labeled 4 are best, while the curves labeled 1 are best in Figures 5(c)′ and 5(d)′. Looking closely at the PH quintic interpolants, we see that the simple -shaped curve with the best shape when becomes less and less acceptable as the parametric speeds at the end-points increase. But the interpolants labeled 1 in Figures 5(a)′, 5(b)′, 5(c)′, and 5(d)′ exhibit the opposite behavior: initially these curves are -shaped loops with high bending energies when ; but as the parametric speed increases, they become -shaped simple curves with lower bending energies. When reaches 20, it has the best shape but the greatest arc-length. This suggests that the best-shaped interpolants, produced by Möbius transformations of PH cubics are more stable than the corresponding PH quintics, in the sense that the former largely achieve a lower arc-length and bending energy than the latter, except when the end-point speeds are significantly asymmetric, as we see when in this example.
Möbius transformations preserve Pythagorean-hodograph properties. For any Hermite data-set, we can generally obtain four Hermite interpolants as Möbius transformations of PH cubics. We have proved that these interpolants are always simple curves or single loops, and that at least two of them must be simple. We have also presented the condition that an interpolant must meet if it is to be a simple curve.
We compared the shape characteristics of Hermite interpolants, produced by Möbius transformations of PH cubics, together with their response to changes of parametric speed at their end points, with the same data for PH quintic interpolants satisfying an identical Hermite dataset: we found that interpolants produced by Möbius transformations of PH cubics generally have lower bending energies and shorter arc-lengths than PH quintics.
One avenue for further research is to look for ways of predicting how the geometry of Möbius transformation of PH cubics will be determined by a particular Hermite data-set. Another avenue to explore would be the application of Möbius transformations to other interpolation problems involving PH (or MPH) curves, in both two and three dimensions. In particular, we might look to complete the geometric characterization of Möbius transformation of PH cubics in Hermite interpolation.
This research was supported by the Korea Science and Engineering Foundation (KOSEF) Grant funded by the Korea government (MEST) (2009-0073488).
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