Geometria Diferencial do conjunto focal
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Data
2018-02-08
Autores
Santos, Samuel Paulino dos
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Universidade Estadual Paulista (Unesp)
Resumo
Seja S uma superf´ıcie regular em R3 sem pontos parab´olicos. O conjunto focal de S ´e o lugar geom´etrico dos centros das esferas que possuem contato degenerado com S em cada ponto. Tal contato ´e medido pelas singularidades da fam´ılia de func¸˜oes distaˆncia ao quadrado D associada a` S. O conjunto focal ´e uma superf´ıcie, por´em n˜ao necessariamente regular, e pode tamb´em ser visto como o conjunto bifurca¸ca˜o da fam´ılia D. A t´ecnica de associar uma variedade singular X(S) a uma subvariedade suave S do espa¸co euclidiano e descobrir alguns aspectos da geometria de S a partir daqueles de X(S) esta´ na essˆencia das aplica¸co˜es da Teoria das Singularidades `a Geometria Diferencial. Neste trabalho, estudamos os modelos, a menos de difeomorfismos, para o conjunto focal de superf´ıcies imersas em R3 gen´ericas, reunimos os principais resultados sobre a geometria da superf´ıcie focal encontrados na literatura e os apresentamos de forma mais explicativa e com uma linguagem moderna. Al´em disso, mostramos que a superf´ıcie focal pode ser parametrizada por uma frente de onda e utilizamos resultados conhecidos para tais aplica¸co˜es no estudo da geometria da superf´ıcie focal.
Let S be a immersed surface in R3 without parabolic points. The focal set of S is the locus of the centres of spheres that have a degenerate contact with S in each point. This contact is measured by singularities of the family of distance squared function D associated with S. The focal set is a surface, but is not necessarily regular, and it can also be seen as the bifurcation set of the family D. The approach of associating a singular variety X(S) to a smooth submanifold S in an Euclidian space and recover some aspects of the geometry of S from that of X(S) is at the essence of applications of singularity theory to the Differential Geometry. In this work, we study models, unless diffeomorphism, of focal set of the immersed generics surfaces in R3. We have also gathered some results about the geometry of the focal set of the literature and we present them in a more explanatory way and in a modern notation. Furthermore, we show that the focal surface can be parametrized by a wave front and use the known results of such applications in the study of the focal set.
Let S be a immersed surface in R3 without parabolic points. The focal set of S is the locus of the centres of spheres that have a degenerate contact with S in each point. This contact is measured by singularities of the family of distance squared function D associated with S. The focal set is a surface, but is not necessarily regular, and it can also be seen as the bifurcation set of the family D. The approach of associating a singular variety X(S) to a smooth submanifold S in an Euclidian space and recover some aspects of the geometry of S from that of X(S) is at the essence of applications of singularity theory to the Differential Geometry. In this work, we study models, unless diffeomorphism, of focal set of the immersed generics surfaces in R3. We have also gathered some results about the geometry of the focal set of the literature and we present them in a more explanatory way and in a modern notation. Furthermore, we show that the focal surface can be parametrized by a wave front and use the known results of such applications in the study of the focal set.
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Palavras-chave
Contato, Desdobramentos versais, Conjunto focal, Frentes de onda, Contact, Versal unfoldings, Focal set, Wave front