Contact detection between a small ellipsoid and another quadric

[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given

Enumerating the morphologies of non-degenerate Darboux cyclides

International audienceWe provide an enumeration of all possible morphologies of non-degenerate Darboux cyclides. Based on the fact that every Darboux cyclide in R 3 is the stereographic projection of the intersection surface of a sphere and a quadric in R 4 , we transform the enumeration problem of morphologies of Darboux cyclides to the enumeration of the algebraic sequences that characterize the intersection of a sphere and a quadric in R 4

Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

International audienceArrangements of geometric objects refer to the spatial partitions formed by the objects and they serve as an underlining structure of motion design, analysis and planning in CAD/CAM, robotics, molecular modeling, manufacturing and computer-assisted radio-surgery. Arrangements are especially useful to collision detection, which is a key task in various applications such as computer animation , virtual reality, computer games, robotics, CAD/CAM and computational physics. Ellipsoids are commonly used as bounding volumes in approximating complex geometric objects in collision detection. In this paper we present an in-depth study on the arrangements formed by two ellipsoids. Specifically, we present a classification of these arrangements and propose an efficient algorithm for determining the arrangement formed by any particular pair of ellipsoids. A stratification diagram is also established to show the connections among all the arrangements formed by two ellipsoids. Our results for the first time elucidate all possible relative positions between two arbitrary ellipsoids and provides an efficient and robust algorithm for determining the relative position of any two given ellipsoids, therefore providing the necessary foundation for developing practical and trustworthy methods for processing ellipsoids for collision analysis or simulation in various applications

Curvas cónicas e superfícies geradas pelas curvas cónicas : os seus traçados geométricos e aplicações no design

No âmbito do Design, é graficamente, sistematiza-se e aprofunda-se o conhecimento das curvas cónicas e, em especial, sobre as suas construções geométricas, contribuindo para o reconhecimento da importância do estudo destas curvas e das superfícies geradas a partir delas no ensino da geometria no Design. Utiliza-se um método expositivo de análise crítica do conhecimento existente, propondo-se ângulos de abordagem menos usuais. Procura-se, ainda, suprir o insuficiente conhecimento e divulgação científica das cónicas em Portugal, e em particular no Design, de três modos: pela sistematização do conhecimento existente, pela adaptação do conhecimento da geometria projetiva, com expressão na geometria plana e na geometria analítica, para a linguagem dos traçados geométricos em geometria plana e em geometria descritiva e, ainda, pela contribuição para a utilização mais generalizada, designadamente com os meios tecnológicos atuais. Paralelamente, identificam-se potencialidades das curvas na capacidade de resolução de problemas de representação gráfica rigorosa, por designers e outros profissionais, designadamente os das áreas das artes visuais e da arquitetura, com reflexos no ensino e na prática profissional. Exemplificam-se utilizações dos traçados das curvas cónicas e das superfícies geradas com elas, no Design e em outras áreas, simplificando soluções e reafirmando a importância e atualidade da geometria plana e da geometria descritiva, tanto no processo de construção do conhecimento e no desenvolvimento do projeto, como nas aplicações práticas, enquanto suporte conceptual e representação gráfica. Tendo em vista a simplificação da aplicação das cónicas e das superfícies geradas com elas em projetos de Design, resolveram-se alguns problemas com solução complexa, ou sem solução, na literatura consultada. Para tal, utilizámos métodos menos usuais, designadamente alguns derivados da geometria descritiva e, sobretudo, procurando integrar os conhecimentos mais recentes sobre as cónicas. Tal abordagem permitiu ainda o aprofundamento de conhecimento com potencial interesse teórico em diversas áreas e o enunciar de algumas propriedades das cónicas que não se encontraram descritas na literatura, numa relação dialética entre teoria e prática, num contexto que contribui para a reafirmação da geometria descritiva, na sua capacidade de desenvolvimento do pensamento criativo.Abstract : In context of Design, and graphically, this research aims to systematize and to deepen the knowledge about the conical curves, and especially on their geometrical constructions, contributing to the recognition of the importance of the study of theses curves and of the surfaces generated from them, mainly in the teaching of geometry in Design courses. An expositive methodology of critical analysis of literature is used, proposing less usual approaches. It also aims to compensate insufficient knowledge and scientific dissemination of conical curves in Portugal, particularly in Design, in three ways: systematizing the existent knowledge, adapting knowledge of projective geometry in plane and analytical geometry for the language of geometrical constructions in plane geometry and descriptive geometry and, still, contributing to a more generalized use involving recent technology. Parallel, the study identifies potentialities of the conical curves in solving problems of rigorous graphic representation for designers and other professionals, namely those from the areas of visual arts and of architecture, with reflexes in the teaching and in the professional practice. It gives examples of practical uses of graphic resolutions of conical curves and of their generated surfaces in Design and other areas, reassuring the importance and relevance of plane geometry and descriptive geometry, both in the construction of knowledge and in the development of the project and in practical applications, as a conceptual framework and graphic representation. Within the scope of simplifying the use of conics and surfaces generated with them in Design projects, are presented solutions for some problems with complex solutions, or without solution, in reviewed literature. The study approach contributes not only to deepen the theoretical frame in several areas but also enunciates some proprieties of conics not mentioned yet in literature, in a dialectical relationship between theory and practice, in a context that contributes to the reaffirmation of descriptive geometry and its capacity of development of creative thinking