Vértices, curva focal e superfície focal de curvas no espaço

• Ano de defesa: 2013
• Autor(a) principal: Wolf, Carla Andreia
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Santa Maria; BR; Matemática; UFSM; Programa de Pós-Graduação em Matemática
• Palavras-chave em Português: Vértices; Cáustica; Curva focal; Singularidades; Vertex; Caustic; Focal curve; Singularities; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.ufsm.br/handle/1/9980

The focal surface of a curve γ in the Euclidean 3-space is defined as the envelope of the normal planes of γ. The focal surface of γ is singular along a curve Cγ, called the focal curve or generalized evolute. This curve is given by the centers of the osculating spheres of γ. In this work we study the geometry of the focal surface, focusing on the properties of the focal curve. These concepts can be generalized for curves in Rm+1. The focal curve may be parametrized in terms of the Frenet frame of γ. Through this parametrization, we obtain coefficients called focal curvatures. It is then obtained a formula relating the Euclidean curvatures of γ with its focal curvatures. Defining a vertex of a curve in Rm+1 as a point at which the curve has at least (m+3)-point contact with its osculating hypersphere, we give necessary and sufficient conditions for a point of γ to be a vertex. In such points the focal surface is locally diffeomorphic to the swallowtail surface.