Computers, Materials & Continua DOI:10.32604/cmc.2021.016006 | |

Article |

Scattered Data Interpolation Using Cubic Trigonometric Bézier Triangular Patch

1Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM, Bangi, 43600, Selangor Darul Ehsan, Malaysia

2Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Seri Iskandar, 32610, Perak Darul Ridzuan, Malaysia

3Department of Fundamental and Applied Sciences and Centre for Systems Engineering (CSE), Institute of Autonoumous System Universiti Teknologi PETRONAS, Seri Iskandar, 32610, Perak Darul Ridzuan, Malaysia

4Faculty of Science, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410, Brunei Darussalam

5Department of Mathematics, Cankaya University, Ankara, Turkey

6Institute of Space Sciences, Magurele-Bucharest, Romania

7Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

*Corresponding Author: Samsul Ariffin Abdul Karim. Email: samsul_ariffin@utp.edu.my

Received: 18 December 2020; Accepted: 18 January 2021

Abstract: This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with

Keywords: Cubic trigonometric; Bézier triangular patches; C1sufficient condition; scattered data interpolation

This paper investigates scattered data interpolation using trigonometric Bézier triangular patch that has been proposed by Zhu et al. [1]. Scattered data interpolation is about the construction of a smooth surface for non-uniform set of data. It can be prescribed by a given a set of scattered data

In a previous study, Saaban et al. [2] performed scattered data interpolation by using minimized sum of squares of principal curvatures. In additions, this scheme also uses geometric continuity which is

Butt et al. [3] proposed a scheme which exhibits the shape preserving properties by positivity, monotonicity and convexity 2D data by inserting more knots in the interval. The positivity of regular data arranged over a rectangular grid was discussed. Hussain et al. [4] proposed

Han [5] proposed cubic trigonometric polynomial curves with shape parameter where the order of continuity is dependent upon the knot vector (uniform or non-uniform) and the value of shape parameters. This scheme shows that the proposed scheme is closer to the control polygon than the corresponding B-spline curves. Besides that, the degree of the cubic trigonometric polynomial curves can be reduced to quadratic trigonometric polynomial curves which represent the ellipse.

Butt [6] preserved the shape of positive data by deriving sufficient conditions for the first partial derivatives and twist values by using a piecewise bi-cubic interpolant. Lamberti et al. [7] also proposed a method for the construction of C2 interpolating function. This scheme preserved the shape of curve via tension parameters. The calculation for approximation order and numerical examples is shown.

Floater [8,9] proposed another shape preserving property which is the convexity where [8] shows derivation of sufficient conditions convexity of tensor-product Bézier surfaces. The conditions focused on

The aim of this paper is to apply scattered data interpolation with trigonometric function which is cubic trigonometric Bézier. To our knowledge, this is the first study that applies trigonometric Bézier triangular for scattered data interpolation. We summarize the main advantages of the proposed scheme as follows:

a) The proposed scattered data interpolation uses cubic trigonometric Bézier with three parameters meanwhile Ali et al. [10], Draman et al. [11] and Karim et al. [12] have used different types of rational interpolants.

b) Our scheme only needs to triangulate the data one time. Meanwhile, Powell–Sabin (PS) and Clough–Tocher (CT) schemes needed to split the macro triangles into several micro triangles for each triangle. This will increase computation time to construct the final interpolating surface.

This paper is organized as follows: Section 2 discusses trigonometric Bézier triangular patches with three shape parameters. Section 3 states the properties of cubic trigonometric Bézier. Section 4 discusses the scattered data interpolation. Section 5 presents the numerical results including comparison with existing schemes. Conclusion and future work are given in Section 6.

2 Trigonometric Bézier Triangular Patch with Three Shape Parameters

Trigonometric Bézier triangular patches is constructed by Zhu et al. [1]. The trigonometric Bézier triangular patches are defined as follows:

Definition 1. Let

the trigonometric Bézier-Like patch over triangular domain with three exponential shape parameters

Noted that,

Definition 2. Let

Properties of Cubic Trigonometric Bézier Triangular Patches

From the definition of the basis function of trigonometric triangular patches, the list below is important properties of the basis [1].

a) Affine invariance and convex hull. The basis function have the properties of partition of unity and nonnegativity, so its simply corresponding that cubic trigonometric Bézier has

b) Geometric property at the corner points. Direct computation such as

c) Corner points tangent plane.

d) Boundary property.

e) Shape adjustable property.

3 Scattered Data Interpolation

In this section, we will discuss the constrution of a smooth surface for given a set of scattered data

Local scheme

This scheme comprises of the convex combination of three local schemes

or

where the local scheme

For inner ordinates, we have employed the cubic precision that was proposed by Foley et al. [15] while Goodman et al. [14] methods are used to calculate the boundary ordinates for each triangle. The vertices

Let the directional derivatives along

Then, applying directional derivative into (3), yields

Other directional derivatives along

Now, we need to calculate the inner ordinates for each triangle. In order to calculate inner ordinates

The inner ordinate

Meanwhile, inner ordinate

The remaining inner ordinates are obtained by symmetry [11].

Now, we establish the algorithm that can be used for surface reconstruction using the proposed scheme.

In this subsection, we discuss the performance of our proposed method by measuring 36,65 and 100 data points. Besides that, we also compare the maximum error, root mean square error (RMSE) and coefficient determination (

• Franke’s exponential function.

where

• Saddle function

• Cliff function

• Gentle function

Fig. 2 shows Delaunay Triangulation of 36, 65 and 100 data points with domain of

Tabs. 1–3 shows numerical result for error measurement for 36, 65 and 100 data points.

Tabs. 1–3 show numerical results for 36, 65 and 100 data points. We can see in Tabs. 1–3, the proposed scheme is on par with two established schemes i.e., Goodman et al. [14] and Karim et al. [16].

Now, we compare the performance between the proposed scattered data interpolation scheme against two well-known scattered data interpolation methods i.e.,

Our final example in this study is to apply the proposed scheme to visualize real scattered data obtained from Ali et al. [19] and Gilat [20]. The electric potential V around a charged particle is given by:

where

where

Fig. 7 shows the Delaunay triangulation for electric potential, and Fig. 8 shows surface interpolant using the proposed scheme.

This paper discusses scattered data interpolation by using cubic trigonometric Bézier triangular patches initiated by Zhu et al. [1]. Sufficient condition for C1 continuity on each adjacent triangle is developed by using cubic precision method. An efficient algorithm is presented. We test the proposed scheme by using four well-known tested functions. We compare the performance against some established schemes such as Goodman et al. [14], Karim et al. [16] and Powell–Sabin (PS) and Clough–Tocher (CT) split schemes. From error analysis, we found that the proposed scheme is on par and for all data sets, we achieve higher R2 values. Finally, we test the proposed scheme to interpolate real scattered data set. For future research, we can apply the proposed scheme for shape preserving interpolation such as positivity and convexity. The proposed scheme also can be applied for constrained surface modeling above, below or between two planes as discussed in Karim et al. [21].

Funding Statement: This research was fully supported by Universiti Teknologi PETRONAS (UTP) and Ministry of Education, Malaysia through research grant FRGS/ 1/2018/STG06/UTP/03/1/015 MA0-020 (New rational quartic spline interpolation for image refinement) and UTP through a research grant YUTP: 0153AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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