THE DUAL BASIS fUNCTIONS FOR CURVES AND SURFACES
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Date
2002-05
Authors
MIOR OTHMAN, WAN AINUN
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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
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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.
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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.
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THE DUAL BASIS fUNCTIONS FOR CURVES AND SURFACES