An in depth analysis, via resultants, of the singularities of a parametric curve
Identifiers
Permanent link (URI): http://hdl.handle.net/10017/41542DOI: 10.1016/j.cagd.2018.12.003
ISSN: 0167-8396
Publisher
Elsevier
Date
2019-01-01Academic Departments
Universidad de Alcalá. Departamento de Física y Matemáticas
Teaching unit
Unidad Docente Matemáticas
Funders
Agencia Estatal de Investigación
Bibliographic citation
Blasco, Ángel & Pérez-Díaz, Sonia. 2019, “An in depth analysis, via resultants, of the singularities of a parametric curve”, Computer Aided Geometric Design, vol. 68 (Enero 2019), pp. 22-47
Keywords
Rational parametrization
Singularities of an algebraic curve
Multiplicity of a point
Ordinary and non-ordinary singularities
T-function
Fiber function
Project
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-88796-P/ES/COMPUTACION SIMBOLICA: NUEVOS RETOS EN ALGEBRA Y GEOMETRIA Y SUS APLICACIONES/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/acceptedVersion
Publisher's version
https://doi.org/10.1016/j.cagd.2018.12.003Rights
Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
© 2019 Elsevier
Access rights
info:eu-repo/semantics/openAccess
Abstract
Let C be an algebraic space curve defined by a rational parametrization P(t)∈K(t)ℓ, ℓ≥2. In this paper, we consider the T-function, T(s), which is a polynomial constructed from P(t) by means of a univariate resultant, and we show that T(s) contains essential information concerning the singularities of C. More precisely, we prove that T(s)=∏i=1nHPi(s), where Pi, i=1,…,n, are the (ordinary and non-ordinary) singularities of C and HPi, i=1,…,n, are polynomials, each of them associated to a singularity, whose factors are the fiber functions of those singularities as well as those other belonging to their corresponding neighborhoods. That is, HQ(s)=HQ(s)m−1∏j=1kHQj(s)mj−1, where Q is an m-fold point, Qj,j=1,…,k, are the neighboring singularities of Q, and mj,j=1,…,k, are their corresponding multiplicities (HP denotes the fiber function of P). Thus, by just analyzing the factorization of T, we can obtain all the singularities (ordinary and non-ordinary) as well as interesting data relative to each of them, like its multiplicity, character, fiber or number of associated tangents. Furthermore, in the case of non-ordinary singularities, we can easily get the corresponding number of local branches and delta invariant.
Files in this item
Files | Size | Format |
|
---|---|---|---|
An_in_depth_Blasco_Comput_Aide ... | 990.0Kb |
|
Files | Size | Format |
|
---|---|---|---|
An_in_depth_Blasco_Comput_Aide ... | 990.0Kb |
|
Collections
- Física y Matemáticas [330]