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Lenimar Nunes De Andrade - One of the best experts on this subject based on the ideXlab platform.

  • Traços de interseção de superficies regulares com passos circulares
    [s.n.], 2018
    Co-Authors: Lenimar Nunes De Andrade
    Abstract:

    Orientador: Wu Shin-TingTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Neste trabalho apresentamos uma técnica mista para o cálculo da interseção de duas superfícies regulares. Nossa técnica consiste em uma variação da técnica da subdivisão dos domínios combinada com trechos de caminhada. Através da subdivisão obtemos pontos próximos da interseção distribuídos aleatoriamente por todo o domínio da parametrização. Selecionamos alguns deles e iniciamos trechos de caminhada usando o que denominamos passo circular. Para uma maior precisão numérica, a caminhada usa em cada ponto uma construção de um círculo osculador aproximado. Para avaliar nossa técnica, fizemos comparações com as técnicas de caminhada já existentes. Baseado nos testes que fizemos podemos afirmar que nossa técnica mista é eficienteAbstract: In this research we present a mixed technique for determining the intersection between two regular surfaces. Our technique is a variation of the domain subdivision technique combined with marching technique. The domain subdivision gives us approximated initial points distributing ramdomly on the .parametrization domain. We choose some of these points and march along the curve using the so-called circular step. In order to get a better numerical precision, we proposed the construction of an approximated Osculating Circle to each point of the curve to determine circular steps. For evaluating our technique, we compared the proposed technique with the existing ones. According to these tests we can assert that our mixed technique is efficientDoutoradoModelagem GeometricaDoutor em Engenharia Elétric

  • Traços de interseção de superficies regulares com passos circulares
    2017
    Co-Authors: Lenimar Nunes De Andrade
    Abstract:

    Resumo: Neste trabalho apresentamos uma técnica mista para o cálculo da interseção de duas superfícies regulares. Nossa técnica consiste em uma variação da técnica da subdivisão dos domínios combinada com trechos de caminhada. Através da subdivisão obtemos pontos próximos da interseção distribuídos aleatoriamente por todo o domínio da parametrização. Selecionamos alguns deles e iniciamos trechos de caminhada usando o que denominamos passo circular. Para uma maior precisão numérica, a caminhada usa em cada ponto uma construção de um círculo osculador aproximado. Para avaliar nossa técnica, fizemos comparações com as técnicas de caminhada já existentes. Baseado nos testes que fizemos podemos afirmar que nossa técnica mista é eficienteAbstract: In this research we present a mixed technique for determining the intersection between two regular surfaces. Our technique is a variation of the domain subdivision technique combined with marching technique. The domain subdivision gives us approximated initial points distributing ramdomly on the .parametrization domain. We choose some of these points and march along the curve using the so-called circular step. In order to get a better numerical precision, we proposed the construction of an approximated Osculating Circle to each point of the curve to determine circular steps. For evaluating our technique, we compared the proposed technique with the existing ones. According to these tests we can assert that our mixed technique is efficien

  • Traços de interseção de superficies regulares com passos circulares
    Universidade Estadual de Campinas . Faculdade de Engenharia Elétrica e de Computação, 1998
    Co-Authors: Lenimar Nunes De Andrade
    Abstract:

    Neste trabalho apresentamos uma técnica mista para o cálculo da interseção de duas superfícies regulares. Nossa técnica consiste em uma variação da técnica da subdivisão dos domínios combinada com trechos de caminhada. Através da subdivisão obtemos pontos próximos da interseção distribuídos aleatoriamente por todo o domínio da parametrização. Selecionamos alguns deles e iniciamos trechos de caminhada usando o que denominamos passo circular. Para uma maior precisão numérica, a caminhada usa em cada ponto uma construção de um círculo osculador aproximado. Para avaliar nossa técnica, fizemos comparações com as técnicas de caminhada já existentes. Baseado nos testes que fizemos podemos afirmar que nossa técnica mista é eficienteIn this research we present a mixed technique for determining the intersection between two regular surfaces. Our technique is a variation of the domain subdivision technique combined with marching technique. The domain subdivision gives us approximated initial points distributing ramdomly on the .parametrization domain. We choose some of these points and march along the curve using the so-called circular step. In order to get a better numerical precision, we proposed the construction of an approximated Osculating Circle to each point of the curve to determine circular steps. For evaluating our technique, we compared the proposed technique with the existing ones. According to these tests we can assert that our mixed technique is efficien

Smutny Jana - One of the best experts on this subject based on the ideXlab platform.

  • Global radii of curvature, and the biarc approximation of space curves:in pursuit of ideal knot shapes
    Lausanne EPFL, 2005
    Co-Authors: Smutny Jana
    Abstract:

    The distance from self-intersection of a (smooth and either closed or infinite) curve q in three dimensions can be characterised via the global radius of curvature at q(s), which is defined as the smallest possible radius amongst all Circles passing through the given point and any two other points on the curve. The minimum value of the global radius of curvature along the curve gives a convenient measure of curve thickness or normal injectivity radius. Given the utility of the construction inherent to global curvature, it is natural to consider variants defined in related ways. The first part of the thesis considers all possible circular and spherical distance functions and the associated, single argument, global radius of curvature functions that are constructed by minimisation over all but one argument. It is shown that among all possible global radius of curvature functions there are only five independent ones. And amongst these five there are two particularly useful ones for characterising thickness of a curve. We investigate the geometry of how these two functions, ρpt and ρtp, can be achieved. Properties and interrelations of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices. It is known that any Lipschitz continuous curve with positive thickness actually has C1,1-regularity. Accordingly, C1,1 is the natural space in which to carry out computations involving self-avoiding curves. The second part of the thesis develops the mathematical theory of biarcs, which are a geometrically elegant way of discretizing C1,1 space curves. A biarc is a pair of circular arcs joined in a C1 fashion according to certain matching rules. We establish a self-contained theory of the geometry of biarc interpolation of point-tangent data sampled from an underlying base curve, and demonstrate that such biarc curves have attractive convergence properties in both a pointwise and function-space sense, e.g. the two arcs of the biarc interpolating a coalescent point-tangent data pair on a C2-curve approach the Osculating Circle of the curve at the limit of the data points, and for a C1,1-base curve and a sequence of (possibly non-uniform) meshes, the interpolating biarc curves approach the base curve in the C1-norm. For smoother base curves, stronger convergence can be obtained, e.g. interpolating biarc curves approach a C2 base curve in the C1,1-norm. The third part of the thesis concerns the practical utility of biarcs in computation. It is shown that both the global radius of curvature function ρpt and thickness can be evaluated efficiently (and to an arbitrarily small, prescribed precision) on biarc curves. Moreover, both the notion of a contact set, i.e. the set of points realising thickness, and an approximate contact set can be defined rigorously. The theory is then illustrated with an application to the computation of ideal shapes of knots. Informally ideal knot shapes can be described as the configuration allowing a given knot to be tied with the shortest possible piece of rope of prescribed thickness. The biarc discretization is combined with a simulated annealing code to obtain approximate ideal shapes. These shapes provide rigorous upper bounds for rope length of ideal knots. The approximate contact set and the function ρpt evaluated on the computed shapes allow us to assess closeness of the computations to ideality. The high accuracy of the computations reveal various, previously unrecognized, features of ideal knot shapes

  • Global radii of curvature, and the biarc approximation of space curves: in pursuit of ideal knot shapes
    2005
    Co-Authors: Smutny Jana, Maddocks, John H.
    Abstract:

    The distance from self-intersection of a (smooth and either closed or infinite) curve q in three dimensions can be characterised via the global radius of curvature at q(s), which is defined as the smallest possible radius amongst all Circles passing through the given point and any two other points on the curve. The minimum value of the global radius of curvature along the curve gives a convenient measure of curve thickness or normal injectivity radius. Given the utility of the construction inherent to global curvature, it is natural to consider variants defined in related ways. The first part of the thesis considers all possible circular and spherical distance functions and the associated, single argument, global radius of curvature functions that are constructed by minimisation over all but one argument. It is shown that among all possible global radius of curvature functions there are only five independent ones. And amongst these five there are two particularly useful ones for characterising thickness of a curve. We investigate the geometry of how these two functions, ρpt and ρtp, can be achieved. Properties and interrelations of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices. It is known that any Lipschitz continuous curve with positive thickness actually has C1,1-regularity. Accordingly, C1,1 is the natural space in which to carry out computations involving self-avoiding curves. The second part of the thesis develops the mathematical theory of biarcs, which are a geometrically elegant way of discretizing C1,1 space curves. A biarc is a pair of circular arcs joined in a C1 fashion according to certain matching rules. We establish a self-contained theory of the geometry of biarc interpolation of point-tangent data sampled from an underlying base curve, and demonstrate that such biarc curves have attractive convergence properties in both a pointwise and function-space sense, e.g. the two arcs of the biarc interpolating a coalescent point-tangent data pair on a C2-curve approach the Osculating Circle of the curve at the limit of the data points, and for a C1,1-base curve and a sequence of (possibly non-uniform) meshes, the interpolating biarc curves approach the base curve in the C1-norm. For smoother base curves, stronger convergence can be obtained, e.g. interpolating biarc curves approach a C2 base curve in the C1,1-norm. The third part of the thesis concerns the practical utility of biarcs in computation. It is shown that both the global radius of curvature function ρpt and thickness can be evaluated efficiently (and to an arbitrarily small, prescribed precision) on biarc curves. Moreover, both the notion of a contact set, i.e. the set of points realising thickness, and an approximate contact set can be defined rigorously. The theory is then illustrated with an application to the computation of ideal shapes of knots. Informally ideal knot shapes can be described as the configuration allowing a given knot to be tied with the shortest possible piece of rope of prescribed thickness. The biarc discretization is combined with a simulated annealing code to obtain approximate ideal shapes. These shapes provide rigorous upper bounds for rope length of ideal knots. The approximate contact set and the function ρpt evaluated on the computed shapes allow us to assess closeness of the computations to ideality. The high accuracy of the computations reveal various, previously unrecognized, features of ideal knot shapes.La distance "d'auto-intersection" d'une courbe tridimensionnelle q (lisse, fermée ou infinie) peut se caractériser par le rayon de courbure global au point q(s), à savoir le plus petit de tous les rayons des cercles passant par le point q(s) et deux autres points quelconques de la courbe. Le rayon de courbure global minimal de la courbe est ainsi une mesure de l'épaisseur ou du rayon injectif normal de la courbe. L'utilité d'une telle notion pour la caractérisation des courbes "non-intersectantes" pousse à étendre la définition précédente du rayon de courbure global. La première partie de la thèse envisage ainsi toutes les fonctions de distance circulaires et sphériques et les fonctions de rayon de courbure global associées obtenues par minimisation de ces fonctions de distance. Il est ainsi démontré que parmi toutes les fonctions de rayon de courbure global seules cinq sont indépendantes, dont deux particulièrement utiles pour caractériser l'épaisseur d'une courbe. Nous décrivons les géométries particulières "réalisant" ces deux fonctions ρpt et ρtp. Les propriétés des fonctions de rayon global de courbure et leurs rélations sont illustrées dans les cas simples des ellipses et des helices. Il est connu qu'une courbe continue, lipschitzienne et d'épaisseur positive est automatiquement de régularité C1,1. C1,1 est par conséquent l'espace naturel pour les calculs mettant en jeu des courbes non-intersectantes. La deuxième partie de la thèse présente ainsi la théorie mathématique des bi-arcs, objets géométriques qui permettent de discrétiser de façon élégante les courbes spatiales de régularité C1,1 : Un bi-arc est une paire d'arcs de cercles joints de manière C1 et suivant certaines règles particulières. Nous proposons une théorie autonome de l'interpolation par bi-arcs d'ensembles de couples point-tangente extraits d'une courbe de base. Nous démontrons en particulier que les courbes de bi-arcs possèdent des propriétés de convergence ponctuelles et uniformes intéressantes. Par exemple, dans la limite où deux point-tangentes d'une courbe C2 se rejoignent, nous montrons que les deux arcs du bi-arc qui interpole cette paire point-tangente approchent le cercle osculatoire au point limite consideré. Nous montrons aussi que pour une courbe C1,1 et une séquence de discrétisation (éventuellement non-uniforme) les courbes bi-arc d'interpolation convergent vers la courbe de base avec la norme C1 et que pour des courbes de bases plus lisses des convergences plus fortes peuvent être obtenues : les courbes bi-arcs interpolant une courbe de base C2 convergent vers la courbe de base avec la norme C1,1. La troisième partie de la thèse concerne l'utilité pratique des bi-arcs dans les calculs. Il est démontré que la fonction de rayon de courbure global ρpt et l'épaisseur des courbes bi-arcs peuvent être évaluées de manière efficace et ce à la précision voulue. Nous définissons de plus de façon rigoureuse la notion d'ensemble de contact, c'est-à-dire l'ensemble des paires de point qui "réalisent" l'épaisseur, ainsi que la notion d'ensemble de contact "approximatif". La théorie est illustrée par l'étude des noeuds idéaux : les configurations des noeuds idéaux correspondent, pour noeud donné et une épaisseur donnée, aux configurations de longueur de corde minimale. Une méthode numérique de minimisation basée sur le principe du recuit-simulé et utilisant la discrétisation en bi-arc permet alors d'obtenir des approximations des configurations de noeuds idéaux. Ces configurations approchées fournissent ainsi des "limites supérieures" rigoureuses de la longueur de corde des noeuds idéaux. L'ensemble de contact approximatif et la fonction ρpt évalués à partir de ces configurations permettent alors d'estimer l' "écart à l'idéalité" de nos résultats. La grande précision des calculs permet par ailleurs de mettre en lumière diverses propriétés non observées jusqu'à présent

Costa, Antonio Evandro De Macedo. - One of the best experts on this subject based on the ideXlab platform.

  • Curvatura de cônicas.
    UFCG, 2018
    Co-Authors: Costa, Antonio Evandro De Macedo.
    Abstract:

    Por meio deste trabalho, pretendemos apresentar o conceito de curvatura do ponto de vista algébrico e geométrico centrado no estudo das curvas cônicas (elipse, hipérbole e parábola). Para isso, apresentamos uma abordagem histórica e, em seguida, um estudo sobre curvas planas culminando na dedução das fórmulas de curvatura e em sua interpretação geométrica usando o círculo osculador. Dando ênfase às curvas cônicas, deduzimos as curvaturas da elipse, hipérbole e parábola, em adição, detalhamos um método para encontrar os centros de curvatura dessas cônicas utilizando apenas procedimentos geométricos. Por fim, abordamos também importantes aplicações físicas do conceito de curvatura.Through this work, we intend to present the concept of curvature from the algebraic and geometric point of view focusing on the study of conical curves (ellipse, hyperbolas and parabola). To this end, we present a historical approach and then a study on plane curves, culminating in the deduction of the curvature formulas and in its geometrical interpretation using the Osculating Circle. By emphasizing the conical curves, we derive the ellipse, hyperbolas and parabola curvature, whereas we detail a method for finding the centers of curvatures of these conics using only geometrical procedures. We also address important physical applications of the concept of curvature.Cape

  • Curvatura de cônicas.
    Centro de Ciências e Tecnologia - CCT, 2018
    Co-Authors: Costa, Antonio Evandro De Macedo.
    Abstract:

    Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-11-22T17:09:06Z No. of bitstreams: 1 ANTONIO EVANDRO DE MACEDO COSTA – DISSERTAÇÃO (PPGMAT) 2017.pdf: 7543467 bytes, checksum: d23fb5e1435fa2ea1c78bd6881520e82 (MD5)Made available in DSpace on 2018-11-22T17:09:06Z (GMT). No. of bitstreams: 1 ANTONIO EVANDRO DE MACEDO COSTA – DISSERTAÇÃO (PPGMAT) 2017.pdf: 7543467 bytes, checksum: d23fb5e1435fa2ea1c78bd6881520e82 (MD5) Previous issue date: 2017-08CapesPor meio deste trabalho, pretendemos apresentar o conceito de curvatura do ponto de vista algébrico e geométrico centrado no estudo das curvas cônicas (elipse, hipérbole e parábola). Para isso, apresentamos uma abordagem histórica e, em seguida, um estudo sobre curvas planas culminando na dedução das fórmulas de curvatura e em sua interpretação geométrica usando o círculo osculador. Dando ênfase às curvas cônicas, deduzimos as curvaturas da elipse, hipérbole e parábola, em adição, detalhamos um método para encontrar os centros de curvatura dessas cônicas utilizando apenas procedimentos geométricos. Por fim, abordamos também importantes aplicações físicas do conceito de curvatura.Through this work, we intend to present the concept of curvature from the algebraic and geometric point of view focusing on the study of conical curves (ellipse, hyperbolas and parabola). To this end, we present a historical approach and then a study on plane curves, culminating in the deduction of the curvature formulas and in its geometrical interpretation using the Osculating Circle. By emphasizing the conical curves, we derive the ellipse, hyperbolas and parabola curvature, whereas we detail a method for finding the centers of curvatures of these conics using only geometrical procedures. We also address important physical applications of the concept of curvature

Edward C. Kinzel - One of the best experts on this subject based on the ideXlab platform.

  • Graphical Technique to Locate the Center of Curvature of a Coupler Point Trajectory
    Journal of Mechanical Design, 2004
    Co-Authors: Gordon R. Pennock, Edward C. Kinzel
    Abstract:

    This paper presents an original technique to locate the center of curvature of the path traced by an arbitrary point fixed in the coupler link of a planar four-bar linkage. The method is purely graphical and the center of curvature; i.e., the center of the Osculating Circle, can be located in a direct manner with few geometric constructions. The advantage of this technique, compared to the classical approach using the Euler-Savary equation, is that measurements of angles and distances between points are not required. Also, it is not necessary to locate inflection points or draw the inflection Circle for the instantaneous motion of the coupler link. The technique is based on the concept of a virtual link which is valid up to, and including, the second-order properties of motion of the coupler link. The virtual link is coincident with the path normal to the coupler curve; i.e., the line connecting the coupler point to the velocity pole of the coupler link. The absolute instant center of the virtual link defines the ground pivot for the link and is, therefore, coincident with the center of the Osculating Circle. The authors believe that the graphical approach presented in this paper represents an important contribution to the kinematics literature on the curvature of a point trajectory.

  • A Graphical Technique to Locate the Center of Curvature of a Coupler Point Trajectory
    Volume 2: 28th Biennial Mechanisms and Robotics Conference Parts A and B, 2004
    Co-Authors: Gordon R. Pennock, Edward C. Kinzel
    Abstract:

    This paper presents an original technique to locate the center of curvature of the path traced by an arbitrary point fixed in the coupler link of a planar four-bar linkage. The method is purely graphical and the center of curvature; i.e., the center of the Osculating Circle, can be located in a direct manner with few geometric constructions. The advantage of this technique, compared to the classical approach using the Euler-Savary equation, is that measurements of angles and distances between points are not required. Also, it is not necessary to locate inflection points or draw the inflection Circle for the instantaneous motion of the coupler link. The technique is based on the novel concept of a virtual link, which is valid up to, and including, the second-order properties of motion of the coupler link. The virtual link is coincident with the path normal to the coupler curve; i.e., the line connecting the coupler point to the velocity pole of the coupler link. The absolute instant center of the virtual link defines the ground pivot for the link and is, therefore, coincident with the center of the Osculating Circle. The authors believe that the graphical technique presented in this paper represents an important contribution to the kinematics literature on the curvature of a point trajectory.Copyright © 2004 by ASME

V. Timorin - One of the best experts on this subject based on the ideXlab platform.

  • Osculating curves: around the Tait-Kneser Theorem
    2016
    Co-Authors: E. Ghys, S. Tabachnikov, V. Timorin
    Abstract:

    The notion of Osculating Circle (or Circle of curvature) of a smooth plane curve is familiar to every student of calculus and elementary differential geometry: this is the Circle that approximates the curve at a point better than all other Circles. One may say that the Osculating Circle passes through three infinitesimally close points on the curve. More specifically, pick three points on the curve and draw a Circle through these points. As the points tend to each other, there is a limiting position of the Circle: this is the Osculating Circle. Its radius is the radius of curvature of the curve, and the reciprocal of the radius is the curvature of the curve. If both the curve and the Osculating Circle are represented locally as graphs of smooth functions then not only the values of these functions but also their first and second derivatives coincide at the point of contact. Ask your mathematical friend to sketch an arc of a curve and a few Osculating Circles. Chances are, you will see something like Figure 1. Figure 1: Osculating Circles? This is wrong! The following theorem was discovered by Peter Guthrie Tait in the end of the 19th century [9] and rediscovered by Adolf Kneser early in the 20th century [4]. Theorem 1 The Osculating Circles of an arc with monotonic positive curvature are pairwise disjoint and nested. Tait’s paper is so short that we quote it almost verbatim (omitting some old-fashioned terms): 1 a

  • Osculating curves: around the Tait-Kneser Theorem
    2014
    Co-Authors: E. Ghys, S. Tabachnikov, V. Timorin
    Abstract:

    The notion of Osculating Circle (or Circle of curvature) of a smooth plane curve is familiar to every student of calculus and elementary differential geometry: this is the Circle that approximates the curve at a point better than all other Circles. One may say that the Osculating Circle passes through three infinitesimally close points on the curve. More specifically, pick three points on the curve and draw a Circle through these points. As the points tend to each other, there is a limiting position of the Circle: this is the Osculating Circle. Its radius is the radius of curvature of the curve, and the reciprocal of the radius is the curvature of the curve. If both the curve and the Osculating Circle are represented locally as graphs of smooth functions then not only the values of these functions but also their first and second derivatives coincide at the point of contact. Ask your mathematical friend to sketch an arc of a curve and a few Osculating Circles. Chances are, you will see something like Figure 1. Figure 1: Osculating Circles? This is wrong! The following theorem was discovered by Peter Guthrie Tait in the end of the 19th century [9] and rediscovered by Adolf Kneser early in the 20th century [4]. Theorem 1 The Osculating Circles of an arc with monotonic positive curvature are pairwise disjoint and nested. Tait’s paper is so short that we quote it almost verbatim (omitting some old-fashioned terms): 1 When the curvature of a plane curve continuously increases or diminishes (as in the case with logarithmic spiral for instance) no two of the Circles of curvature can intersect each other. This curious remark occurred to me some time ago in connection with an accidental feature of a totally different question... The proof is excessively simple. For if A, B, be any two points of the evolute, the chord AB is the distance between the centers of two of the Circles, and is necessarily less than the arc AB, the difference of their radii... When the curve has points of maximum or minimum curvature, there are corresponding... cusps on the evolute; and pairs of Circles of curvature whose centers lie on opposite sides of the cusp, C, may intersect: – for the chord AB may now exceed the difference between CA and CB. See Figure 2 for a family of Osculating Circles of a spiral.