Hermite Interpolation with PH Curves Using the Enneper Surface

Hindawi Abstract and Applied Analysis Volume 2018, Article ID 5680723, 10 pages https://doi.org/10.1155/2018/5680723 Research Article 𝐶1 Hermite Interpolation with PH Curves Using the Enneper Surface Hyun Chol Lee,1 Jae Hoon Kong,1 and Gwangil Kim 1 1,2 Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea 2 Correspondence should be addressed to Gwangil Kim; 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. We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regular 𝐶1 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 R2 with polynomial speed functions, which have polynomial arclengths, 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 R𝑛 [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 R𝑚,1 [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 𝐺1 Hermite [20, 22, 23] and 𝐶2 Hermite conditions [7]. In particular, 𝐶1 Hermite interpolation problems have been solved by several techniques [13, 24–27] including PHpreserving 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 𝐶1 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 𝐶1 Hermite interpolation with PH curves, there have been many developments: in particular, it has been shown [24] that 𝐶1 Hermite interpolation problems with PH curves in R3 can be reduced to problems in R2 and generic interpolants can then be obtained to satisfy a given 𝐶1 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 𝐶1 Hermite data-sets. We will address both of these issues: using the Enneper surface, we can solve 𝐶1 Hermite interpolation problems more efficiently for all regular 𝐶1 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 PHpreserving mapping and give examples. In Section 3, we show that the parametrization of the Enneper surface in standard 2 Abstract and Applied Analysis form is PH-preserving and that, by rescaling the Enneper surface, we can find two cubic surfaces that satisfy any regular 𝐶1 Hermite data-set. We also prove that we can obtain eight interpolants on the two cubic surfaces that satisfy a regular 𝐶1 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. In addition, let P(𝑢, V) = (𝑥(𝑢, V), 𝑦(𝑢, V), 𝑧(𝑢, V)) be a polynomial mapping given by 𝑥 (𝑢, V) = 𝑢5 − 10𝑢3 V2 + 5𝑢V4 + 𝑢3 − 3𝑢V2 , 𝑦 (𝑢, V) = V5 − 10V3 𝑢2 + 5V𝑢4 + V3 − 3V𝑢2 , 𝑧 (𝑢, V) = 2√15𝑢V (𝑢2 − V2 ) . Then, for a r(𝑡) = (𝑢(𝑡), V(𝑡)) in R2 , since 𝑑 𝑥 (𝑢 (𝑡) , V (𝑡)) = 5𝑢 (𝑡)4 𝑢󸀠 (𝑡) − 30𝑢 (𝑡)2 V (𝑡)2 𝑢󸀠 (𝑡) 𝑑𝑡 − 20𝑢 (𝑡)3 V (𝑡) V󸀠 (𝑡) 2. Preliminary + 5V (𝑡)4 𝑢󸀠 (𝑡) 𝑛 Let R be the 𝑛-dimensional Euclidean space, for 𝑛 ∈ N, and let P[𝑡] be the set of polynomial functions with real coefficients. We express a polynomial curve in R𝑛 as a mapping r : R → R𝑛 from the space of real numbers R to R𝑛 , such that the component functions of r, which are 𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡), are members of P[𝑡]. Definition 1. A polynomial curve r(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡)) is said to be a Pythagorean-hodograph (PH) curve if its velocity vector or hodograph r󸀠 (𝑡) = (𝑥1󸀠 (𝑡), . . . , 𝑥𝑛󸀠 (𝑡)) satisfies the Pythagorean condition 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩r (𝑡)󵄩󵄩 = 𝑥1󸀠 (𝑡)2 + 𝑥2󸀠 (𝑡)2 + ⋅ ⋅ ⋅ + 𝑥𝑛󸀠 (𝑡)2 = 𝜎 (𝑡)2 , 󵄩 󵄩𝑛 + 20𝑢 (𝑡) V (𝑡) V (𝑡) + 3𝑢 (𝑡)2 𝑢󸀠 (𝑡) − 3V (𝑡)2 𝑢󸀠 (𝑡) − 6𝑢 (𝑡) V (𝑡) V󸀠 (𝑡) , 𝑑 𝑦 (𝑢 (𝑡) , V (𝑡)) = 5V (𝑡)4 V󸀠 (𝑡) − 30V (𝑡)2 𝑢 (𝑡)2 V󸀠 (𝑡) 𝑑𝑡 − 20V (𝑡)3 𝑢 (𝑡) 𝑢󸀠 (𝑡) + 5𝑢 (𝑡)4 V󸀠 (𝑡) 3 + 3V (𝑡)2 V󸀠 (𝑡) − 3𝑢 (𝑡)2 V󸀠 (𝑡) − 6𝑢 (𝑡) V (𝑡) 𝑢󸀠 (𝑡) , 𝑑 𝑧 (𝑢 (𝑡) , V (𝑡)) = 6√15𝑢 (𝑡)2 V (𝑡) 𝑢󸀠 (𝑡) 𝑑𝑡 Example 3. Let Ψ be an affine transformation given by for ∀p ∈ R3 , + 2√15𝑢 (𝑡)3 V󸀠 (𝑡) (2) where R is an orthogonal matrix in R3 , 𝜆 ∈ R \ {0} is a scaling factor, and k is a constant vector in R3 . Then, for a PH curve ̃ : R2 → R3 defined r(𝑡) = (𝑢(𝑡), V(𝑡)) in R2 , the mapping Ψ by ̃ (r (𝑡)) = Ψ (p (𝑡)) , Ψ where p (𝑡) = (𝑢 (𝑡) , V (𝑡) , 0) (3) is PH-preserving, since 2󵄩 󵄩 󸀠 󵄩2 = 𝜆 ⟨p (𝑡) , p (𝑡)⟩ = 𝜆 󵄩󵄩󵄩p (𝑡)󵄩󵄩󵄩󵄩3 󸀠 󸀠 = 𝜆2 (𝑢󸀠 (𝑡)2 + V󸀠 (𝑡)2 ) , where ⟨, ⟩ denotes the usual inner product in R3 . we obtain 2 𝑑 󵄩2 󵄩󵄩 󵄩󵄩P (r (𝑡))󸀠 󵄩󵄩󵄩 = ( 𝑥 (𝑢 (𝑡) , V (𝑡))) 󵄩3 󵄩 𝑑𝑡 2 2 𝑑 𝑑 𝑦 (𝑢 (𝑡) , V (𝑡))) + ( 𝑧 (𝑢 (𝑡) , V (𝑡))) 𝑑𝑡 𝑑𝑡 2 = ⟨𝜆Rp (𝑡) , 𝜆Rp (𝑡)⟩ 2 − 6√15𝑢 (𝑡) V (𝑡)2 V󸀠 (𝑡) , = (𝑢 (𝑡)2 + V (𝑡)2 ) (5𝑢 (𝑡)2 + 5V (𝑡 (...truncated)