Abstract
We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.
Highlights
Implicitization algorithms have been studied in both the CAGD and algebraic geometry communities for many years
For CAGD systems based on floating point arithmetic, exact implicitization methods are often unfeasible due to performance issues
The methods we present attempt to find “best fit” implicit curves or surfaces of a given degree m the definition of Journal of Applied Mathematics
Summary
Implicitization algorithms have been studied in both the CAGD and algebraic geometry communities for many years. Approximate methods that are well suited to floating-point implementation have emerged in the past 25 years 2–6. These methods are closely related to the algorithms we present; those that fit most closely into the framework of this paper include 4, 7– 9. Exact implicit representations of rational parametric manifolds often have very high polynomial degrees, which can cause numerical instabilities and slow floating-point calculations. For CAGD systems based on floating point arithmetic, exact implicitization methods are often unfeasible due to performance issues. The methods we present attempt to find “best fit” implicit curves or surfaces of a given degree m the definition of Journal of Applied Mathematics “best fit” varies with regard to the chosen method of approximation.
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