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

^{1}sufficient condition

This paper investigates scattered data interpolation using trigonometric Bézier triangular patch that has been proposed by Zhu et al. [

In a previous study, Saaban et al. [

Butt et al. [

Han [

Butt [^{2} interpolating function. This scheme preserved the shape of curve

Floater [^{2} and smaller SMSE and maximum error, however, their scheme took longer computational time to generate the results. Meanwhile Draman et al. [

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:

The proposed scattered data interpolation uses cubic trigonometric Bézier with three parameters meanwhile Ali et al. [

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.

Trigonometric Bézier triangular patches is constructed by Zhu et al. [

Noted that,

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

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

Geometric property at the corner points. Direct computation such as

Corner points tangent plane.

Boundary property.

Shape adjustable property.

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. [

Let the directional derivatives along

Then, applying directional derivative into

From

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 [

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

1) Input data points |

2) Triangulate the data sites using Delaunay triangulation method. |

3) Derivation |

4) Generate the surfaces using cubic trigonometric triangular patches |

5) Compute the error–maximum error, RMSE and R^{2} |

6) Compare the performances with two existing method–cubic Ball and cubic Bézier. |

7) Repeat 1 until 6 using different test function. |

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

Now, we compare the performance between the proposed scattered data interpolation scheme against two well-known scattered data interpolation methods ^{2}.

Input data points Data are triangulated using Delaunay triangulation method. Estimate the gradients at the vertices of the triangulation from the scattered data for PS and CT Schemes. Compute the interpolants and generate the surfaces Calculate the error–maximum error, RMSE and R Repeat 1 until 6 using different test function. |

Test function | Method | Shape parameter | Max error | RMSE | |||
---|---|---|---|---|---|---|---|

Cubic ball | 0.1122 | 0.0264 | 0.9915 | ||||

Cubic Bézier | 0.1051 | 0.0263 | 0.9916 | ||||

Cubic trigonometric Bézier | 4 | 3 | 4 | 0.0975 | 0.0275 | 0.9908 | |

5 | 3 | 4.5 | 0.0930 | 0.0278 | 0.9906 | ||

6 | 5 | 4.5 | 0.0984 | 0.0288 | 0.9899 | ||

Cubic ball | 0.0265 | 0.0060 | 0.9936 | ||||

Cubic Bézier | 0.0262 | 0.0061 | 0.9934 | ||||

Cubic trigonometric Bézier | 3 | 3 | 4 | 0.0262 | 0.0062 | 0.9944 | |

3 | 4 | 5 | 0.0263 | 0.0068 | 0.9928 | ||

2 | 4 | 5 | 0.0262 | 0.0068 | 0.9930 | ||

Cubic ball | 0.0491 | 0.0130 | 0.9829 | ||||

Cubic Bézier | 0.0483 | 0.0129 | 0.9832 | ||||

Cubic trigonometric Bézier | 2 | 2.5 | 4 | 0.0531 | 0.0132 | 0.9825 | |

3 | 2 | 4 | 0.0526 | 0.0131 | 0.9826 | ||

3 | 2 | 5 | 0.0535 | 0.0131 | 0.9837 | ||

Cubic ball | 0.0127 | 0.0041 | 0.9973 | ||||

Cubic Bézier | 0.0103 | 0.0037 | 0.9978 | ||||

Cubic trigonometric Bézier | 2 | 2 | 2 | 0.0119 | 0.0042 | 0.9973 | |

3 | 2 | 2 | 0.0123 | 0.0040 | 0.9982 | ||

2 | 2 | 3 | 0.0132 | 0.0042 | 0.9972 |

Our final example in this study is to apply the proposed scheme to visualize real scattered data obtained from Ali et al. [

where

where

Test function | Method | Shape parameter | Max error | RMSE | |||
---|---|---|---|---|---|---|---|

Cubic ball | 0.0611 | 0.0154 | 0.9971 | ||||

Cubic Bézier | 0.0643 | 0.0152 | 0.9972 | ||||

Cubic trigonometric Bézier | 2 | 4.5 | 4 | 0.0625 | 0.0153 | 0.9977 | |

2 | 4 | 5 | 0.0627 | 0.0157 | 0.9970 | ||

3 | 6 | 5 | 0.0631 | 0.0162 | 0.9968 | ||

Cubic ball | 0.0130 | 0.0031 | 0.9983 | ||||

Cubic Bézier | 0.0153 | 0.0033 | 0.9981 | ||||

Cubic trigonometric Bézier | 4 | 3 | 4 | 0.0178 | 0.0039 | 0.9973 | |

3 | 2 | 3 | 0.0164 | 0.0037 | 0.9987 | ||

3 | 3 | 3 | 0.0181 | 0.0037 | 0.9976 | ||

Cubic ball | 0.0309 | 0.0049 | 0.9976 | ||||

Cubic Bézier | 0.0312 | 0.0049 | 0.9975 | ||||

Cubic trigonometric Bézier | 2 | 2 | 2 | 0.0327 | 0.0054 | 0.9971 | |

2 | 2 | 3 | 0.0318 | 0.0050 | 0.9978 | ||

3 | 2 | 2 | 0.0327 | 0.0054 | 0.9970 | ||

Cubic ball | 0.0072 | 0.0020 | 0.9994 | ||||

Cubic Bézier | 0.0060 | 0.0018 | 0.9995 | ||||

Cubic trigonometric Bézier | 2 | 2 | 2 | 0.0075 | 0.0020 | 0.9995 | |

2 | 3 | 3 | 0.0087 | 0.0023 | 0.9991 | ||

2 | 2 | 3 | 0.0082 | 0.0023 | 0.9992 |

Test |
Method | Shape parameter | Max |
RMSE | |||
---|---|---|---|---|---|---|---|

Cubic ball | 0.0336 | 0.0067 | 0.9995 | ||||

Cubic Bézier | 0.0342 | 0.0070 | 0.9994 | ||||

Cubic trigonometric Bézier | 3 | 3 | 3 | 0.0362 | 0.0082 | 0.9992 | |

2 | 3 | 2 | 0.0352 | 0.0080 | 0.9992 | ||

3 | 3 | 2 | 0.0351 | 0.0080 | 0.9995 | ||

Cubic ball | 0.0057 | 0.0013 | 0.9997 | ||||

Cubic Bézier | 0.0046 | 0.0011 | 0.9998 | ||||

Cubic trigonometric Bézier | 4 | 3 | 3 | 0.0057 | 0.0015 | 0.9996 | |

4 | 3.5 | 4 | 0.0067 | 0.0012 | 0.9997 | ||

5 | 3 | 3 | 0.0064 | 0.0017 | 0.9995 | ||

Cubic ball | 0.0226 | 0.0035 | 0.9988 | ||||

Cubic Bézier | 0.0238 | 0.0035 | 0.9988 | ||||

Cubic trigonometric Bézier | 2 | 2 | 2 | 0.0280 | 0.0038 | 0.9985 | |

3 | 2 | 2 | 0.0275 | 0.0038 | 0.9985 | ||

4 | 3 | 2 | 0.0271 | 0.0039 | 0.9985 | ||

Cubic ball | 0.0054 | 0.0010 | 0.9998 | ||||

Cubic Bézier | 0.0034 | 0.0007 | 0.9999 | ||||

Cubic trigonometric Bézier | 2 | 2 | 2 | 0.0039 | 0.0011 | 0.9998 | |

2 | 2 | 3 | 0.0038 | 0.0011 | 0.9998 | ||

2 | 3 | 2 | 0.0041 | 0.0012 | 0.9998 |

Num. of data points | Function | Maximum error (MaxE) | RMSE | ||
---|---|---|---|---|---|

PS interpolant | CT interpolant | PS interpolant | CT interpolant | ||

100 | 1 | 3.40e−02 | 3.41e−02 | 7.12e−03 | 6.48e−03 |

2 | 3.89e−03 | 3.62e−03 | 6.55e−04 | 6.15e−04 | |

3 | 2.02e−02 | 2.08e−02 | 3.84e−03 | 3.75e−03 | |

4 | 7.63e−03 | 6.66e−03 | 1.51e−03 | 1.38e−03 | |

65 | 1 | 9.43e−02 | 1.01e−01 | 1.96e−02 | 1.83e−02 |

2 | 1.91e−02 | 1.82e−02 | 3.27e−03 | 3.23e−03 | |

3 | 2.97e−02 | 2.89e−02 | 6.44e−03 | 5.90e−03 | |

4 | 1.37e−02 | 1.22e−02 | 3.04e−03 | 2.82e−03 | |

36 | 1 | 1.47e−01 | 1.44e−01 | 4.01e−02 | 3.90e−02 |

2 | 3.76e−02 | 3.08e−02 | 7.74e−03 | 7.87e−03 | |

3 | 6.45e−02 | 5.24e−02 | 1.49e−02 | 1.47e−02 | |

4 | 6.10e−02 | 5.34e−02 | 1.26e−02 | 1.20e−02 |

Num. of data points | Function | PS Interpolant | CT Interpolant |
---|---|---|---|

100 | 1 | 0.9994 | 0.9995 |

2 | 0.9999 | 1.0000 | |

3 | 0.9985 | 0.9986 | |

4 | 0.9997 | 0.9997 | |

65 | 1 | 0.9953 | 0.9960 |

2 | 0.9987 | 0.9987 | |

3 | 0.9958 | 0.9965 | |

4 | 0.9987 | 0.9989 | |

36 | 1 | 0.9805 | 0.9822 |

2 | 0.9927 | 0.9930 | |

3 | 0.9775 | 0.9782 | |

4 | 0.9773 | 0.9797 |

This paper discusses scattered data interpolation by using cubic trigonometric Bézier triangular patches initiated by Zhu et al. [^{1} 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. [^{2} 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. [

^{1}scattered data interpolation with minimized sum of squares of principal curvatures

^{1}positivity preserving scattered data interpolation using rational Bernstein–Bézier triangular patch

^{2}functional interpolation via parametric cubics

^{1}triangular interpolant suitable for scattered data interpolation