Inversion, degree, reparametrization and implicitization of improperly parametrized planar curves using μ-basis

We consider the proper reparametrization problem of rational curves by introducing the μ-bases. The results are essential to the theoretical completeness of the theory of μ-bases. We provide an interchange graph for the rational curves that are not necessarily proper (see Figure). The red parts can...

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Bibliographic Details
Published in:Computer aided geometric design Vol. 84; p. 101957
Main Authors: Pérez-Díaz, Sonia, Shen, Li-Yong
Format: Journal Article
Language:English
Published: Elsevier B.V 01.01.2021
Subjects:
Algebraic curves
Fibre
Implicitization
Inversion
Proper reparametrization
Rational parametrization
μ-basis
Implicitization
Proper reparametrization
Rational parametrization
Inversion
Fibre
Algebraic curves
μ-basis
ISSN:0167-8396, 1879-2332
Online Access:Get full text
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Summary:We consider the proper reparametrization problem of rational curves by introducing the μ-bases. The results are essential to the theoretical completeness of the theory of μ-bases. We provide an interchange graph for the rational curves that are not necessarily proper (see Figure). The red parts can be found in previous works while the blue parts are proposed in this paper. •The μ-basis of a planar curve is well studied and some new properties are given.•Two new proper reparametrization methods are presented using μ-basis.•We propose a more complete interchange graph for the representations of rational planar curves. The μ-basis of a rational curve/surface is a new algebraic tool which plays an important role in connecting the rational parametric form and the implicit form of a rational curve/surface. However, most results for μ-bases are presented for proper rational parametrizations. In this paper we consider the μ-basis for an improper rational planar curve. Based on the known properties and new results, we design two new proper reparametrization algorithms using μ-basis. The inversion, degree of the induced rational map and implicitization formulas are also derived.
ISSN:0167-8396
1879-2332
DOI:10.1016/j.cagd.2021.101957