Ball surface representations using partial differential equations

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...

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Main Author: Kherd, Ahmad Saleh Abdullah
Format: Thesis
Language:eng
eng
Published: 2015
Subjects:
QA75 Electronic computers
Computer science
Online Access:https://etd.uum.edu.my/5391/1/s93357.pdf
https://etd.uum.edu.my/5391/2/s93357_abstract.pdf
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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). 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