Ball surface representations using partial differential equations
Over two decades ago, geometric modelling using partial differential equations (PDEs) approach was widely studied in Computer Aided Geometric Design (CAGD). This approach was initially introduced by some researchers to deal with Bèzier surface related to the minimal surface area determined by prescr...
Saved in:
Main Author: | |
---|---|
Format: | Thesis |
Language: | eng eng |
Published: |
2015
|
Subjects: | |
Online Access: | https://etd.uum.edu.my/5391/1/s93357.pdf https://etd.uum.edu.my/5391/2/s93357_abstract.pdf |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
id |
my-uum-etd.5391 |
---|---|
record_format |
uketd_dc |
institution |
Universiti Utara Malaysia |
collection |
UUM ETD |
language |
eng eng |
advisor |
Saaban, Azizan |
topic |
QA75 Electronic computers Computer science |
spellingShingle |
QA75 Electronic computers Computer science Kherd, Ahmad Saleh Abdullah Ball surface representations using partial differential equations |
description |
Over two decades ago, geometric modelling using partial differential equations (PDEs) approach was widely studied in Computer Aided Geometric Design (CAGD). This approach was initially introduced by some researchers to deal with Bèzier surface related to the minimal surface area determined by prescribed boundary curves. However, Bèzier surface representation can be improved in terms of computation time and minimal surface area by employing Ball surface representation. Thus, this research develops
an algorithm to generalise Ball surfaces from boundary curves using elliptic PDEs. Two specific Ball surfaces, namely harmonic and biharmonic, are first constructed in developing the proposed algorithm. The former and later surfaces require two and four boundary conditions respectively. In order to generalise Ball surfaces in the polynomial solution of any fourth order PDEs, the Dirichlet method is then employed. The numerical results obtained on well-known example of data points show that the
proposed generalised Ball surfaces algorithm performs better than BCzier surface representation in terms of computation time and minimal surface area. Moreover, the new constructed algorithm also holds for any surfaces in CAGD including the Bèzier surface. This algorithm is then tested in positivity preserving of surface and image enlargement
problems. The results show that the proposed algorithm is comparable with the existing methods in terms of accuracy. Hence, this new algorithm is a viable alternative for constructing generalized Ball surfaces. The findings of this study contribute towards the body of knowledge for surface reconstruction based on PDEs approach in the area of geometric modelling and computer graphics. |
format |
Thesis |
qualification_name |
Ph.D. |
qualification_level |
Doctorate |
author |
Kherd, Ahmad Saleh Abdullah |
author_facet |
Kherd, Ahmad Saleh Abdullah |
author_sort |
Kherd, Ahmad Saleh Abdullah |
title |
Ball surface representations using partial differential equations |
title_short |
Ball surface representations using partial differential equations |
title_full |
Ball surface representations using partial differential equations |
title_fullStr |
Ball surface representations using partial differential equations |
title_full_unstemmed |
Ball surface representations using partial differential equations |
title_sort |
ball surface representations using partial differential equations |
granting_institution |
Universiti Utara Malaysia |
granting_department |
Awang Had Salleh Graduate School of Arts & Sciences |
publishDate |
2015 |
url |
https://etd.uum.edu.my/5391/1/s93357.pdf https://etd.uum.edu.my/5391/2/s93357_abstract.pdf |
_version_ |
1747827923990085632 |
spelling |
my-uum-etd.53912021-03-18T08:24:43Z Ball surface representations using partial differential equations 2015 Kherd, Ahmad Saleh Abdullah Saaban, Azizan Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA75 Electronic computers. Computer science Over two decades ago, geometric modelling using partial differential equations (PDEs) approach was widely studied in Computer Aided Geometric Design (CAGD). This approach was initially introduced by some researchers to deal with Bèzier surface related to the minimal surface area determined by prescribed boundary curves. However, Bèzier surface representation can be improved in terms of computation time and minimal surface area by employing Ball surface representation. Thus, this research develops an algorithm to generalise Ball surfaces from boundary curves using elliptic PDEs. Two specific Ball surfaces, namely harmonic and biharmonic, are first constructed in developing the proposed algorithm. The former and later surfaces require two and four boundary conditions respectively. In order to generalise Ball surfaces in the polynomial solution of any fourth order PDEs, the Dirichlet method is then employed. The numerical results obtained on well-known example of data points show that the proposed generalised Ball surfaces algorithm performs better than BCzier surface representation in terms of computation time and minimal surface area. Moreover, the new constructed algorithm also holds for any surfaces in CAGD including the Bèzier surface. This algorithm is then tested in positivity preserving of surface and image enlargement problems. The results show that the proposed algorithm is comparable with the existing methods in terms of accuracy. Hence, this new algorithm is a viable alternative for constructing generalized Ball surfaces. The findings of this study contribute towards the body of knowledge for surface reconstruction based on PDEs approach in the area of geometric modelling and computer graphics. 2015 Thesis https://etd.uum.edu.my/5391/ https://etd.uum.edu.my/5391/1/s93357.pdf text eng public https://etd.uum.edu.my/5391/2/s93357_abstract.pdf text eng public Ph.D. doctoral Universiti Utara Malaysia Ahmad, D. & Masud, B. (2014). Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation. Applied Mathematics and Computation, 233,72-84. Aphirukmatakun, C. & Dejdumrong, N. (2007). An approach to polynomial curve comparison in geometric object database. International Journal of Computer Science, 2(4), 240-246. Aphirukmatakun, C. & Dejdumrong, N. (2011). Multiple degree elevation and constrained multiple degree reduction for dp curves and surfaces. Computers & Mathematics with Applications, 61 (8), 2296-2299. Arnal, A. & Monterde, J. (2014). Generating harmonic surfaces for interactive design. Computers & Mathematics with Applications, 67(10), 1914-1924. Arnal, A., Monterde, J. & Ugail, H. (2011). Explicit polynomial solutions of fourth order linear elliptic partial differential equations for boundary based smooth surface generation. Computer Aided Geometric Design, 28(6), 382-394. Bakhshesh, D. & Davoodi, M. (2014). Approximating of conic sections by dp curves with endpoint interpolation. Ball, A. A. (1974). Consurf. part one: introduction of the conic lofting tile. Computer-Aided Design, 6(4), 243-249. Ball, A. A. (1975). Consurf. part two: description of the algorithms. Computer-Aided Design, 7(4), 237-242. Ball, A. A. (1977). Consurf. part three: How the program is used. Computer-Aided Design, 9(1), 9-12. Bloor, M. I. & Wilson, M. J. (1990). Representing pde surfaces in terms of b-splines. Computer-Aided Design, 22(6), 324-331. Chang, G. (1982). Matrix formulations of btzier technique. Computer-aided design, 14(6), 345-350. Chen, X. D., Xua, G. & Wanga, Y. (2009). Approximation methods for the plateaubCzier problem. In Cad/graphics (pp. 588-591). Cosin, C. & Monterde, J. (2002). BCzier surfaces of minimal area. In Computational science-iccs 2002 (pp. 72-81). Springer. Dan, Y. & Xinmeng, C. (2007). Another type of generalized ball curves and surfaces. Acta Mathematica Scientia, 27(4), 897-907. Dejdumrong, N. (2006). Rational dp-ball curves. In Computer graphics, imaging and visualisation, 2006 international conference on (pp. 478483). Dejdurnrong, N. (2011). Monomial form is universal. Journal of Computer and Advanced Technologies. Dejdumrong, N. (2014). Monomial form approach to the construction and the conversion for curves in cagd. Taylor & Francis Group, London. Delgado, J. & Pefia, J. M. (2003a). A linear complexity algorithm for the bernstein basis. In Geometric modeling and graphics, international conference on (pp. 162-162). Delgado, J. & Peiia, J. M. (2003b). A shape preserving representation with an evaluation algorithm of linear complexity. Computer Aided Geometric Design, 20(1), 1-10. Do Carmo, M. & Perdigao. (1976). DiSferential geometry of curves and sur$aces (Vol. 2). Prentice-Hall Englewood Cliffs. Du, H. & Qin, H. (2004). A shape design system using volumetric implicit pdes. Computer-Aided Design, 36(11), 1101-1116. Farin, G. (2002). Curves and surj6ace.s for computer aided geometricdesign [Book]. San Diego: Academic San Diego. Farin, G. & Hansford, D. (1999). Discrete coons patches. Computer Aided Geometric Design, 16(7), 691-700. Faux, I. D. & Pratt, M. J. (1979). Computational geometry for design and manufacture. Ellis Horwood Ltd. Goodman, T. N. & Said, H. (1991). Properties of generalized ball curves and surfaces. Computer-Aided Design, 23(8), 554-560. Han, D. (2013). Comparison of commonly used image interpolation methods. In Proceedings of the 2nd international conference on computer science and electronics engineering. Hu, S., Wang, G. & Sun, J. (1998). A type of triangular ball surface and its properties. Journal of Computer Science and Technology, 13(1), 63-72. Hu, S., Wang, G. Z. & Jin, T. G. (1996). Properties of two types of generalized ball curves. Computer-Aided Design, 28(2), 125-1 33. Itsariyawanich, K. & Dejdurnrong, N. (2008). Degree reduction and multiple degree reduction for the dp curves. In Electrical engineering/electronics, computer; telecommunications and information technology, 2008. ecti-con 2008. 5th international conference on (Vol. 1, pp. 45-48). Jiang, S. R. & Wang, G. J. (2005). Conversion and evaluation for two types of parametric surfaces constructed by ntp bases. Computers & Mathematics with Applications, 49(2), 321-329. Kahyaolu, K. & Emin. (2014). An approach for minimal surface family passing a curve. arXiv preprint arXiv 1408.3723. Kim, C. H., Seong, S. M. & Lee, L. S. (2003). Winscale: an image-scaling algorithm using an area pixel model. Circuits and Systems for Video Technology, IEEE Transactions on, 13(6), 549-553. Li, C. Y., Wang, R. H. & Zhu, C. G. (2013). Designing approximation minimal parametric surfaces with geodesics. Applied Mathematical Modelling, 37(9), 6415-6424. Monterde, J. (2004). BCzier surfaces of minimal area: The dirichlet approach. Computer Aided Geometric Design, 21 (2), 117-136. Monterde, J. & Ugail, H. (2004). On harmonic and biharmonic bCzier surfaces. Computer Aided Geometric Design, 21 (7), 697-715. Monterde, J. & Ugail, H. (2006). A general 4th-order pde method to generate bCzier surfaces from the boundary. Computer Aided Geometric Design, 23(2), 208-225. Phien, H. N. & Dejdumrong, N. (2000). Efficient algorithms for bCzier curves. Computer Aided Geometric Design, 17(3), 247-250. Saaban, A., Man, N. & Karim, S. (2013). Surface interpolation using partial differential equation with positivity preserving cubic bÈzier curves boundary condition. Far East Journal of Mathematical Sciences (FJMS), 75(2), 257-272. Said, H. (1989). A generalized ball curve and its recursive algorithm. ACM Transactions on Graphics (TOG), 8(4), 360-371. Sheng, Y., Sourin, A., Castro, G. G. & Ugail, H. (2010). A pde method for patchwise approximation of large polygon meshes. The Visual Computer, 26(6-8), 975-984. Trisdahl, (3. & Rgnquist, E. M. (2011). High order numerical approximation of minimal surfaces. Journal of Computational Physics, 230(12), 4795-4810. Ugail, H. (201 1). Partial diflerential equations for geometric design. Springer. Ugail, H., Mbrquez, M. & Yilmaz, A. (2011). On bCzier surfaces in three-dimensional minkowski space. Computers & Mathematics with Applications, 62(8), 2899-2912. Wang & Cheng. (2001). New algorithms for evaluating parametric surface. Progress in Natural Science, 11 (2), 142-148. Wang & Guo. (2012). Extending and correcting some research results on minimal and harmonic surfaces. Computer Aided Geometric Design, 29(1), 41-50. Wang, G. (1987). Ball curve of high degree and its geometric properties. Applied Mathematics: A journal of Chinese universities, 2(1), 126-140. Xua, G. & Wang, G. Z. (2010). Quintic parametric polynomial minimal surfaces and their properties. Diferential Geometry and its Applications, 28(6), 697-704. Yang, H. & Wang, G. (2015). Optimized design of bèzier surface through bèzier geodesic quadrilateral. Journal of Computational and Applied Mathematics, 273, 264-273. Zhang, J. & You, L. (2004). Fast surface modelling using a 6th order pde. In Computer graphics forum (Vol. 23, pp. 311-320). Zhanga, J., Caia, P. & Wanga, G. (2011). A new approch to design rational harmonic surface over rectangular or triangular domain. Journal of Information & Computational Science, 8(1), 71-83. |