Abstract and Applied Analysis

Volume 2018, Article ID 5680723, 10 pages

https://doi.org/10.1155/2018/5680723

## Hermite Interpolation with PH Curves Using the Enneper Surface

^{1}Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea^{2}Research 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 [2–4]. 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 [7–9] and their practical applications [7–11]. 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 [17–21], in particular to meet Hermite [20, 22, 23] and Hermite conditions [7]. In particular, Hermite interpolation problems have been solved by several techniques [13, 24–27] 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 .