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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2019.tde-23082019-170529
Document
Author
Full name
Alex Paulo Francisco
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2019
Supervisor
Committee
Tari, Farid (President)
Hernandes, Maria Elenice Rodrigues
Manoel, Miriam Garcia
Martins, Luciana de Fátima
 
Title in Portuguese
Deformações geométricas de curvas no plano Minkowski
Keywords in Portuguese
Curvas planas
Inflexões
Plano Minkowski
Singularidades
Vértices
Abstract in Portuguese
Neste trabalho, estendemos o método desenvolvido em (SALARINOGHABI, 2016),(SALARINOGHABI; TARI, 2017) para curvas no plano Minkowski. Tal método propõe um modo de estudar deformações de curvas planas levando em consideração a geometria das mesmas juntamente com suas singularidades. Abordamos detalhadamente todos os fenômenos locais que ocorrem genericamente em famílias de curvas a 2-parâmetros. Em cada caso, obtemos a geometria da curva deformada, ou seja, informações a respeito de inflexões, vértices e pontos lightlike. Obtemos também o comportamento da evoluta/cáustica de uma curva em pontos especiais e as bifurcações que podem aparecer ao deformá-la. Além disso, a fim de obter as deformações genéricas em uma inflexão lightlike de ordem 2, também classificamos submersões de R3 em R por meio de difeomorfismos na fonte que preservam a swallowtail e, utilizando tal classificação, estudamos a geometria plana da swallowtail, a qual provém de seu contato com planos, o qual por sua vez é medido pelas singularidades da função altura sobre a swallowtail.
 
Title in English
Geometric deformations of curves in the Minkowski plane
Keywords in English
Inflections
Minkowski plane
Plane curves
Singularities
Vertices
Abstract in English
In this work, we extend the method developed in (SALARINOGHABI, 2016),(SALARINOGHABI; TARI, 2017) to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in detail with all local phenomena that occur generically in 2-parameters families of curves. In each case, we obtain the geometry of the deformed curve, that is, information about inflections, vertices and lightlike points. We also obtain the behavior of the evolute/caustic of a curve at special points and the bifurcations that can occur when the curve is deformed. Moreover, in order to obtain the generic deformations at a lightlike inflection point of order 2, we also classify submersions from R3 to R by diffeomorphisms in the source that preserve the swallowtail and, using such classification, we study the flat geometry of the swallowtail, which comes from its contact with planes, which in turn is measured by the singularities of the height function on the swallowtail.
 
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Publishing Date
2019-08-26
 
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