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Abstract and Applied Analysis
Volume 2018, Article ID 5680723, 10 pages
https://doi.org/10.1155/2018/5680723
Research Article

Hermite Interpolation with PH Curves Using the Enneper Surface

1Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
2Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

Correspondence should be addressed to Gwangil Kim; rk.ca.ung@mikig

Received 4 January 2018; Accepted 3 April 2018; Published 8 May 2018

Academic Editor: Khalil Ezzinbi

Copyright © 2018 Hyun Chol 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.

Abstract

We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regular Hermite data-sets. Each such data-set is satisfied by two scaled Enneper surfaces, and we can obtain four interpolants on each surface. Examples of these interpolants were found to be better, in terms of bending energy and arc-length, than those obtained using a previous PH-preserving mapping.

1. Introduction

Pythagorean-hodograph (PH) curves were first introduced by Farouki and Sakkalis [1] as polynomial curves in with polynomial speed functions, which have polynomial arc-lengths, rational curvature functions, and rational offsets, all of which derive from their polynomial speed functions. These properties make PH curves good candidates for CAGD and CAD/CAM applications such as interpolation of discrete data and control of motion along curved paths [24]. Also, these PH curves have subsequently been extended, with several applications, to rational curves with rational speed functions in [3, 5, 6].

PH curves have been the subject of a great deal of study, both their formal representation [79] and their practical applications [711]. PH curves have been generalized [11] to participate in medial axis transforms [3, 12], becoming MPH curves in the Minkowski space [8, 13, 14], and this has motivated a lot of further research. There has also been a lot of work on the use of PH curves for interpolating planar [7, 8, 10, 15, 16] and spatial data-sets [1721], in particular to meet Hermite [20, 22, 23] and Hermite conditions [7]. In particular, Hermite interpolation problems have been solved by several techniques [13, 2427] including PH-preserving mappings [24], which have recently been extended [13] to MPH-preserving mappings.

In this paper, we show that we can use Enneper surfaces to solve Hermite interpolation problems with PH curves, by exploiting two properties of the Enneper surface: the geometric property that it contains two straight lines and the PH-preserving nature of its parametrization. Since Farouki and Neff’s original work on Hermite interpolation with PH curves, there have been many developments: in particular, it has been shown [24] that Hermite interpolation problems with PH curves in can be reduced to problems in and generic interpolants can then be obtained to satisfy a given Hermite data-set. This is achieved by a special cubic PH-preserving mapping which satisfies the data-set. However, significant drawbacks remain with this method: one is that the algebraic manipulations required are long and complicated; and the other is that this method is restricted to a special class of Hermite data-sets. We will address both of these issues: using the Enneper surface, we can solve Hermite interpolation problems more efficiently for all regular Hermite data-sets; and we will show that the interpolants obtained by this method may be expected to have better shapes than those obtained by the special mapping, in terms of both bending energy and arc-length.

The rest of this paper is organized as follows: In Section 2, we define the Pythagorean-hodograph curve and the PH-preserving mapping and give examples. In Section 3, we show that the parametrization of the Enneper surface in standard form is PH-preserving and that, by rescaling the Enneper surface, we can find two cubic surfaces that satisfy any regular Hermite data-set. We also prove that we can obtain eight interpolants on the two cubic surfaces that satisfy a regular Hermite data-set. In Section 4, we compare our method with the use of PH-preserving cubic mappings [24], from two different perspectives: the amount of algebraic computation required and the geometric characterizations of the resulting curves. By empirical comparison of interpolants for the same data-set, we show that our method is more efficient and stable than the use of mappings. In Section 5, we summarize the results of this work and propose some themes for further study.

2. Preliminary

Let be the -dimensional Euclidean space, for , and let be the set of polynomial functions with real coefficients. We express a polynomial curve in as a mapping from the space of real numbers to , such that the component functions of , which are , are members of .

Definition 1. A polynomial curve is said to be a Pythagorean-hodograph (PH) curve if its velocity vector or hodograph satisfies the Pythagorean condition where denotes the Euclidean norm of and .

Definition 2. A polynomial mapping is said to be PH-preserving if, for every PH curve in , is a PH curve in .

Example 3. Let be an affine transformation given by where is an orthogonal matrix in , is a scaling factor, and is a constant vector in . Then, for a PH curve in , the mapping defined by is PH-preserving, since where denotes the usual inner product in .

In addition, let be a polynomial mapping given by Then, for a in , since we obtain This means that is PH-preserving. Figure 1(a) shows the image of in , which is the PH-preserving surface parameterized by . The blue curve on the surface is the image of the curve shown in Figure 1(b), which is a segment of the Tchirnhausen cubic. This is a PH curve in , and thus the curve on the surface is a PH curve in .

Figure 1: (a) A PH-preserving surface parameterized by , derived in Example 3, when ; the blue curve on the surface is the image of , shown in (b).

3. Construction of Hermite PH Interpolants on the Scaled Enneper Surface

Definition 4. Let be a surface with the parametrization given by is called the Enneper surface in standard form.

It is known [28] that PH curves in the domain of the Enneper surface can be mapped to PH curves on the surface: thus the parametrization of the Enneper surface is PH-preserving. We revisit this result briefly.

Theorem 5. is PH-preserving.

Proof. Assume that is a PH curve in . Then, since the curve in is a PH curve. This completes the proof.

Theorem 6. contains two straight lines and on the -plane .

Proof. If , or equivalently or , we immediately obtain or . This completes the proof.

Remark 7. The Enneper surface is the nontrivial polynomial minimal surface of the lowest possible degree; equivalently it is area-minimizing, and the parametrization is conformal (i.e., angle-preserving). Now we consider the surfaces obtained by rescaling the Enneper surface. Let be a polynomial surface of degree given by the parametrization where is the standard parametrization of the Enneper surface and is a nonzero real number. Then is also a polynomial minimal surface of degree , and is PH-preserving, as shown in Example 3. From now on, we will call the surface the scaled Enneper surface (or s-Enneper surface) associated with a scaling factor . Note that all s-Enneper surfaces contain and , which we will use in Theorem 9.

Definition 8. A Hermite data-set consists of two end-points and , and two velocities and at those end-points, where , , and . is said to be regular if , and are linearly independent.

Also note that if and lie on a surface , and and are tangents to that surface, then we can say that satisfies . If is a mapping which generates the surface , then we can also say that satisfies .

Theorem 9. For a regular Hermite data-set , we can find two s-Enneper surfaces satisfying .

Proof. Let be the s-Enneper surface obtained by the parametrization where is the parametrization of the Enneper surface in standard form. First, since for all , it is obvious that lies on . In addition, as stated in Remark 7, by Theorem 5, we can find a point in given by and a suitable scaling factor , such that : since , we can assume the existence of some and such that Hence, if we have . That is, and lie on the s-Enneper surface parameterized by . Next we note that, since and , the unit vector normal to at is . This means that . Moreover, sincewe can obtain the unit vector normal to at as follows:Here, since we can solvewhen and we obtainThen, since the given data-set is regular, it is obvious that . Therefore and are tangent to the s-Enneper surface when Consequently, satisfies . This completes the proof.

Remark 10. Note that, if a data-set is not regular, the three vectors , , and must lie on the same plane. This means that the interpolation problem that must be solved to obtain PH curves satisfying the given data-set is equivalent to a planar Hermite interpolation, which has been studied thoroughly [3, 10, 15, 16], in parallel, including the case of solving the interpolation problem for such a data-set with MPH curves [14, 29, 30].

Example 11. Let . Then, using Theorem 9, we can obtain the two s-Enneper surfaces shown in Figure 2: Figure 2(a) is the s-Enneper surface when and and Figure 2(b) is the s-Enneper surface when and .

Figure 2: Two s-Enneper surfaces satisfying in Example 11: (a) the s-Enneper surface that satisfies when and ; and (b) the s-Enneper surface that satisfies when and .

Theorem 12. For a Hermite data-set , , where . We can obtain four PH interpolants satisfying on each of the two s-Enneper surfaces which satisfy .

Proof. By Theorem 9, for , with , we can obtain two s-Enneper surfaces satisfying . Then, by using the parametrization of each s-Enneper surface, we can obtain a new Hermite data-set on from the given , which satisfies , , , and , where denotes the derivative mapping of . Let and ; then for each parametrization of the s-Enneper surfaces satisfying , we have that and as shown in the proof of Theorem 9. Next, assume that and . Then, since solving simultaneously and with respect to , , and , we obtainFinally, we solve the Hermite interpolation problem for the reduced data-set with planar PH quintics in . Note that if the given data-set is regular, then the reduced data-set is also regular, since is conformal (i.e., angle-preserving). For the regular Hermite data-set , we can always [10] find four interpolants: two of them are simple and the others are loops. Therefore, since is PH-preserving, if is a PH interpolant satisfying , then is also a PH interpolant satisfying the given data-set , which lies on the s-Enneper surface with the parametrization . Consequently, we can obtain four PH interpolants satisfying on each of the two s-Enneper surfaces which satisfy the data-set. Moreover, since is conformal, the topological property of each interpolant on the s-Enneper surface, whether it is a simple curve or a loop, is the same as that of the preimage of itself obtained from the inverse of . This completes the proof.

Example 13. Let be the same Hermite data-set as in Example 11. Then, by Theorem 12, we can obtain eight interpolants on the two s-Enneper surfaces which satisfy , as shown in Figure 3. They consist of four interpolants on the s-Enneper surface , with and , as shown in Figure 3(b), and four interpolants on the s-Enneper surface , with and , as shown in Figure 3(d). These eight interpolants are obtained from the images of the corresponding planar interpolants, shown in Figures 3(a) and 3(c), which satisfy the reduced data-sets obtained by the parametrization of each s-Enneper surface where .

Figure 3: Two s-Enneper surfaces and interpolants satisfying and () in Example 13: (a) four interpolants satisfying ; (b) four interpolants satisfying on ; (c) four interpolants satisfying ; (d) four interpolants satisfying on .

Corollary 14. For an arbitrary regular Hermite data-set in , we can obtain eight interpolants on the surfaces that satisfy , by scaling and exploiting the isometry of the two s-Enneper surfaces.

Proof. Let be an affine mapping in , obtained by composing a translation, an orthogonal transformation, and a scaling, as shown in Example 3, so that and . Then, using , we can obtain a new regular Hermite data-set as follows: where and are, respectively, the scaling factor and the orthogonal matrix of . Here we can use Theorem 12 to obtain eight interpolants on two s-Enneper surfaces satisfying . In addition, as shown in Example 3, since is PH-preserving, we can obtain eight interpolants on the cubic surfaces given by , where is the s-Enneper surface satisfying , for .

Example 15. Let , , , . Then, as stated in Corollary 14, using the orthogonal matrix given by with the scaling factor , we can reduce the given data-set to the Hermite data-set , .

By Theorems 9 and 12, we can obtain two s-Enneper surfaces and eight interpolants satisfying . So, using the inverse transformation for the surfaces and interpolants, we finally obtain two cubic surfaces and interpolants satisfying the original data-set , as shown in Figure 4: The cubic surface and the interpolants shown in Figure 4(b) are obtained by the inverse transformation , where is the s-Enneper surface obtained when and ; and the cubic surface and the interpolants in Figure 4(d) are obtained by , where is the s-Enneper surface obtained when and .

Figure 4: The s-Enneper surfaces and interpolants that satisfy in Example 15: (a) four planar interpolants when and ; (b) four interpolants satisfying on the cubic surface ; (c) four planar interpolants when and ; (d) four interpolants satisfying on the cubic surface .

Remark 16. Note that when the given data-set is not regular, that is, , and are linearly dependent, these three vectors must lie on a plane in . That allows us to reduce this interpolation problem to a planar Hermite interpolation problem using a suitable PH-preserving affine transformation. Thus we can obtain [10] four planar PH interpolants satisfying .

4. Comparison with Interpolants Obtained by Other PH-Preserving Mappings

In this section, we compare our method with the use of PH-preserving cubic mappings [24]. First note that, for a regular Hermite data-set in , we can use mappings to obtain interpolants satisfying , which consist of four interpolants on each of four cubic surfaces satisfying , as shown in Figure 5(a). Whereas, applying our method to the same data-set, we can obtain eight interpolants satisfying , which consist of four interpolants on each of two cubic surfaces satisfying , as shown in Figure 5(b). Now we will compare these two methods in terms of their requirements for algebraic computation and the geometries of the resulting curves.

Figure 5: Comparison of interpolants satisfying of 11 in [24]: (a) 16 interpolants obtained by using general PH-preserving cubic mappings; (b) 8 interpolants obtained using the s-Enneper surface. In each subfigure, the interpolant with the lowest bending energy is shown as the orange curve; the shortest interpolant is shown as the green curve; and the longest interpolant is shown as the curve denoted by dashed lines.

Our method requires less algebraic computation to determine the cubic surfaces satisfying the given data-set than the mapping method: when using the mapping method, to determine the PH-preserving surfaces satisfying the given data-set, lengthy computation processes are additionally required in fixing the free parameters of the surfaces, since they have many free parameters with complicated constraints unlike Enneper surfaces. Moreover, our method only requires the given data-set to meet the regularity condition, whereas mapping requires a data-set which meets several conditions.

We will now examine the shapes of the interpolants produced by mapping and by our method. Looking at the interpolants in Figure 5, it is clear that those in Figure 5(a) are longer and more complicated than those in Figure 5(b), which means that the surfaces satisfying the given data-set must be different.

Next, we will compare the interpolants obtained using each method in terms of bending energy: where and are the curvature and torsion of an interpolant . The bending energy of a curve is an established measure of its fairness. We can consider an interpolant to have a better shape than another if it has lower bending energy with a similar arc-length. If we look at the bending energies and arc-lengths in Tables 1 and 2, we observe the following:(i)While the curve with the lowest bending energy in Figure 5(a) has the longest arc-length, which is about twice that of the shortest curve, the curve with the lowest bending energy in Figure 5(b) is not much longer than the shortest curve.(ii)The diversity of arc-lengths among the curves in Figure 5(a) is much larger than the diversity of those in Figure 5(b), where the lengths of the shortest and longest curves are in the ratio 1 : 2.36.(iii)Finally, the lowest bending energy of any curve in Figure 5(b) is 63% lower than that of the curve with the lowest bending energy in Figure 5(a).

Table 1: Comparison of the bending energies and arc-length of the 3 important interpolants shown in Figure 5(a).
Table 2: Comparison of the bending energies and arc-lengths of the 3 important interpolants shown in Figure 5(b).

These results suggest that our method can produce better interpolants through a more convenient and shorter algebraic computation than mapping.

5. Concluding Remarks and Suggestions for Further Study

We have proved that the parametrization of the Enneper surface in standard form is PH-preserving and that it also preserves the - and -axes in the parametric plane. We went on to show how to produce interpolants which lie on two s-Enneper surfaces and satisfy a regular Hermite data-set in . We also proved that eight interpolants can be obtained, four on each of the surfaces. We also compared our method with a previous method [24] based on PH-preserving cubic mappings and showed that we can obtain interpolants with lower bending energy without significant increase in arc-length.

The work reported in this paper raises further questions: first, as shown in Section 4, by using different parameterizations to solve the Hermite interpolation problem satisfying a single data-set , we obtain different cubic surfaces. Then the Enneper surface is a minimal surface: does that mean that all the cubic PH-preserving mappings are harmonic? Further, are the surfaces produced by the cubic PH-preserving mappings used in the mapping method also minimal? This brings us to the question of how to characterize all possible cubic PH-preserving mappings. Now, as shown in 3 of [24], PH-preserving mappings are not necessarily harmonic; and so PH-preserving surfaces need not to be minimal.

Extending our perspective to include PH-preserving mappings raises another question which brings us nearer to core of the PH-preserving property: what is the key to the property of PH preservation: conformality, harmonicity, or both? finally, we might consider the following interesting questions: can we design new PH-preserving mappings by using specific PH curves in such as PH-cuts of Laurent series [27]? We have been considering these last two questions and currently believe that the PH-preserving property of mappings is mainly dependent upon conformality. We have to be in a position to write on this topic shortly. In addition, PH and MPH curves are tightly connected, as stated in [11]. Hence, considering our recent work [13] on MPH-preserving mappings, we also naturally reach the following questions: what would be the equivalent of Enneper surfaces in the Minkowski setting? Can we again achieve interpolation, but this time with MPH curves? These could also be the nice themes for further study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1E1A1A03070952).

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