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Um estudo sobre curvas, superfícies e suas parametrizações
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Este trabalho foi desenvolvido com o intuito de estudar as propriedades locais de curvas e superfícies no plano e espaço euclidianos, expressando-as parametricamente. Procurou-se descrever a forma da curva através da curvatura e da torção. Seguidamente, foram analisados vários métodos de obter novas curvas através de uma curva inicial, nomeadamente a evoluta, involuta, pedal, cissóide e conchóide. No que respeita a uma superfície, a sua forma é descrita através das curvaturas gaussiana e média. Adicionalmente, sobre as curvas contidas em superfícies, a sua forma é descrita através das curvaturas normal, geodésica e principal. As geodésicas são abordadas como sendo curvas contidas numa superfície, apurando a menor distância entre dois pontos na superfície. Saliente-se ainda que, devido à dificuldade e escassez de visualização de tais parametrizações na literatura, pretendeu-se ao longo do trabalho exemplificar graficamente cada caso.
This work was developed with the aim to study the local properties of curves and surfaces, in the euclidean space, through their parametrizations. We described the shape of a curve through its curvature and torsion. Next, we analysed several methods of obtaining new curves from an original curve, namely the evolute, involute, pedal, cissoid and conchoid of a curve. Concerning surfaces, its shape is described by Gaussian and mean curvatures. Additionally, concerning curves contained in surfaces, its shape is described through normal, geodesic and principal curvatures. Geodesics are approached as curves contained in a surface, minimizing the distance between two points on the surface. It should also be pointed out that due to the difficulty and scarcity of visualization of such parametrizations in the literature, it was intended throughout this work to exemplify each case graphically.
This work was developed with the aim to study the local properties of curves and surfaces, in the euclidean space, through their parametrizations. We described the shape of a curve through its curvature and torsion. Next, we analysed several methods of obtaining new curves from an original curve, namely the evolute, involute, pedal, cissoid and conchoid of a curve. Concerning surfaces, its shape is described by Gaussian and mean curvatures. Additionally, concerning curves contained in surfaces, its shape is described through normal, geodesic and principal curvatures. Geodesics are approached as curves contained in a surface, minimizing the distance between two points on the surface. It should also be pointed out that due to the difficulty and scarcity of visualization of such parametrizations in the literature, it was intended throughout this work to exemplify each case graphically.
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Curvas Curvatura Equações de Frenet Geodésica Geometria diferencial Superfícies Curves Curvature Differential Geometry Frenet equations Geodesic Surfaces Matemática . Faculdade de Ciências Exatas e da Engenharia