THE DUAL BASIS fUNCTIONS FOR CURVES AND SURFACES
| dc.contributor.author | MIOR OTHMAN, WAN AINUN | |
| dc.date.accessioned | 2016-01-12T03:51:19Z | |
| dc.date.available | 2016-01-12T03:51:19Z | |
| dc.date.issued | 2002-05 | |
| dc.description.abstract | A fundamental problem in Computer Aided Geometric Design (CAGD) is defining, representing and manipulating curves and surfaces when these objects have to be processed by a computer. Evaluation, subdivision, knot insertion, knot deletion interpolation, degree elevation and differentiation are some of the many algorithms developed to solve this problem. These algorithms can be viewed simply as a change of basis procedures. This thesis introduces the dual basis functions as a new tool to develop change of basis procedures between different polynomial and spline bases associated with the respective curves and surfaces. The dual basis functions are derived by using techniques analogous to the construction of the de Boor-Fix dual functionals and applying Marsden's Identity. We found that various formulae and theorems remain valid when the basis is replaced by its dual basis functions. Also, in some cases the dual basis functions can be used to represent the dual functionals of the primary basis. The dual basis functions for the nezier curves and tensor product and triangular Bezier surfaces are derived and used to prove that degree elevation <'(nd differentiation for Bernstein basis are closely interrelated. An explicit formula for the dual basis functions to the Generalized Ball basis of odd degree is also derived. These functions enable us to derive some important explicit formulae of the Generalized Ball basis which could not be developed otherwise. A xi change of basis formula from Bezier basis to Ball basis and a subdivision formula for the Generalized Ball curves are also derived. These formulae are easy to adopt and require less computations than other recursive procedures. We note that our subdivision formula is not numerically ill conditioned as in other methods. We also derive an explicit formula for the dual basis functions of the bivariate Generalized Ball basis and used the dual basis functions to derive the change of basis algorithms from the triangular Bezier basis to the triangular Generalized Ball basis. Next, an explicit formula for the dual basis functions of the rational Bezier basis is derived. We showed that these dual functions can be expressed as linear combinations of Bernstein polynomials. Finally, the explicit formula for the dual basis funtion for box splines over the three directional mesh is defined. .. ~ We note here that all the formulae for the dual basis functions that we derived are defined explicitly, and could easily be used as a tool for change of basis procedures, as compared to the recursive forms. | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/1484 | |
| dc.subject | THE DUAL BASIS fUNCTIONS FOR CURVES AND SURFACES | en_US |
| dc.title | THE DUAL BASIS fUNCTIONS FOR CURVES AND SURFACES | en_US |
| dc.type | Thesis | en_US |
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