NEW CUBIC TIMMER TRIANGULAR PATCHES WITH C¹ AND G¹ CONTINUITY

Authors

  • Fatin Amani Mohd Ali Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak Darul Ridzuan, Malaysia
  • Samsul Ariffin Abdul Karim Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak Darul Ridzuan, Malaysia Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia
  • Sarat Chandra Dass Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak Darul Ridzuan, Malaysia
  • Vaclav Skala School of Computer Science and Engineering, University of West Bohemia, Plzen
  • Azizan Saaban School of Quantitative Sciences, UUMCAS, Universiti Utara Malaysia, Kedah, Malaysia
  • Mohammad Khatim Hasan Centre for Artificial Intelligence Technology, Faculty of Information Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
  • Ishak Hashim Centre for Modelling & Data Science, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

DOI:

https://doi.org/10.11113/jt.v81.13759

Keywords:

Cubic Timmer triangular patch, Parametric continuity, Geometric continuity, cubic Timmer curve, scattered

Abstract

In this study, a new cubic Timmer triangular patch is constructed by extending the univariate cubic Timmer basis functions. The best scheme that lies towards the control polygon is cubic Timmer curve and surface compared to the other methods. From the best of our knowledge, nobody has extended the univariate cubic Timmer basis to the bivariate triangular patch. The construction of the proposed cubic Timmer triangular patch is based on the main idea of the cubic Ball and cubic Bezier triangular patches construction. Some properties of the new cubic Timmer triangular patch are investigated. Furthermore, the composite cubic Timmer triangular patches with parametric continuity (C1) and geometric continuity (G1) are discussed. Simple error analysis between the triangular patches and one test function is provided for each continuity type. Numerical and graphical results are presented by using Mathematica and MATLAB.

Author Biographies

  • Fatin Amani Mohd Ali, Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak Darul Ridzuan, Malaysia
    Fundamental and Applied Sciences Department
  • Samsul Ariffin Abdul Karim, Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak Darul Ridzuan, Malaysia Centre for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia
    Fundamental and Applied Sciences Department
  • Sarat Chandra Dass, Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak Darul Ridzuan, Malaysia
    Fundamental and Applied Sciences Department
  • Vaclav Skala, School of Computer Science and Engineering, University of West Bohemia, Plzen
    School of Computer Science and Engineering
  • Azizan Saaban, School of Quantitative Sciences, UUMCAS, Universiti Utara Malaysia, Kedah, Malaysia
    School of Quantitative Sciences
  • Mohammad Khatim Hasan, Centre for Artificial Intelligence Technology, Faculty of Information Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
    Centre for Artificial Intelligence Technology, Faculty of Information Science and Technology
  • Ishak Hashim, Centre for Modelling & Data Science, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
    Centre for Modelling & Data Science, Faculty of Science & Technology

References

Ali, J. M. 1994. An Alternative Derivation of Said Basis Functions. Sains Malaysiana. 23(3): 42-56.

Ali, F. A. M., Karim, S. A. A., Dass, S. C., Skala, V., Saaban, A., Hasan, M. K., and Ishak, H. 2019. Efficient Visualization of Scattered Energy Distribution Data by Using Cubic Timmer Triangular Patches. To be published by Springer.

Awang, N., and Rahmat, R. W. 2017. Reconstruction of Smooth Surface by Using Cubic Bezier Triangular Patch in GUI. Malaysian Journal of Industrial Technology. 2(1).

Awang, N., Rahmat, R. W., Sulaiman, P. S., and Jaafar, A. 2017. Delaunay Triangulation of a Missing Points. Journal of Advanced Science and Engineering. 7(1): 58-69.

Brodlie, K., Mashwama, P., & Butt, S. 1995. Visualization of Surface Data to Preserve Positivity and other Simple Constraints. Computers & Graphics. 19(4): 585-594.

Chan, E. S., & Ong, B. H. 2001. Range Restricted Scattered Data Interpolation Using Convex Combination of Cubic Bézier Triangles. Journal of Computational and Applied Mathematics. 136(1-2): 135-147.

Chang, L. H. T., & Said, H. B. 1997. A C2 Triangular Patch for the Interpolation of Functional Scattered Data. Computer-Aided Design. 29(6): 407-412.

Farin, G. 1986. Triangular Bernstein-bézier Patches. Computer Aided Geometric Design. 3(2): 83-127.

Farin, G. 2014. Curves and Surfaces for Computer-aided Geometric Design: A Practical Guide. Elsevier.

Foley, T. A., & Opitz, K. 1992. Hybrid Cubic Bézier Triangle Patches. Mathematical Methods in Computer Aided Geometric Design II. 275-286.

Goodman, T. N. T., & Said, H. B. 199). A C1 Triangular Interpolant Suitable for Scattered Data Interpolation. Communications in Applied Numerical Methods. 7(6): 479-485.

Karim, S. A. A., and Saaban, A. 2018. Visualization Terrain Data Using Cubic Ball Triangular Patches. MATEC Web of Conferences 225. 06023.

Liu, C. 2001. Theory and Application of Convex Curves and Surfaces in CAGD. Faculty of Mathematical Sciences, University of Twente.

Luo, Z., & Peng, X. 2006. A C1-rational Spline in Range Restricted Interpolation of Scattered Data. Journal of Computational and Applied Mathematics. 194(2): 255-266.

Mann, S. 2000. Continuity Adjustments to Triangular Bézier Patches that Retain Polynomial Precision. Polar. 1(2): 2.

Ramli, N., and Ali, J. M. 2014. Object Design Using Blending of Rational Timmer. AIP Conference Proceedings. 1605(1): 262-267.

Ong, B. H., & Wong, H. C. 1996. A C1 Positivity Preserving Scattered Data Interpolation Scheme. Series in Approximations and Decompositions. 8: 259-274.

Saaban, A., Majid, A. A., Piah, M., & Rahni, A. 2009. Visualization of Rainfall Data Distribution Using Quintic Triangular Bézier Patches. Bulletin of the Malaysian Mathematical Sciences Society. 32(2).

Said, H. B. 1990. The Bezier Ball Type Cubic Curves and Surfaces. Sains Malaysia. 19(4): 85-95.

Said, H. B., & Wirza, R. 1993. A Cubic Ball Triangular Patch for the Scattered Data Interpolation. Pusat Pengajian Sains Matematik dan Sains komputer, Universiti Sains Malaysia.

Shepard, D. 1985. A two Diemnsional Interpolation Function for Irregularly Spaced Data. Proc. ACM Nat. Conf. 517-524.

Timmer, H. G. 1980. Alternative Representation for Parametric Cubic Curves and Surfaces. Computer Aided Design. 12: 25-28.

Wu, J., Zhang, X. and Peng, L. 2010. Positive Approximation and Interpolation Using Compactly Supported Radial Basis Functions. Mathematical Problems in Engineering. Article ID 964528, 10 pages.

Wu, J, Lai, Y. and Zhang, X. 2010. Radial Basis Functions for Shape Preserving Planar Interpolating Curves. Journal of Information & Computational Science. 7(7): 1453-1458.

Zhu, Y., Han, X., & Liu, S. 2014. Quartic Rational Said-Ball-like Basis with Tension Shape Parameters and Its Application. Journal of Applied Mathematics.

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Published

2019-09-22

Issue

Vol. 81 No. 6: November 2019

Section

Science and Engineering

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Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.

How to Cite

NEW CUBIC TIMMER TRIANGULAR PATCHES WITH C¹ AND G¹ CONTINUITY. (2019). Jurnal Teknologi, 81(6). https://doi.org/10.11113/jt.v81.13759