Квадратичные оптимизационные задачи компьютерной геометрии

Работа посвящена постановке и решению класса квадратичных оптимизационных задач компьютерной геометрии: поиск эллипсоида минимального объема, содержащего множество точек евклидового пространства, поиск минимального расстояния между эллипсоидами, построение гиперплоскости, разделяющей два эллипсоида. Предложены эффективные алгоритмы для решения этого класса задач.Робота присвячена постановці та розв’язку класу квадратичних оптимізаційних задач комп’ютерної геометрії: пошук еліпсоїду мінімального об’єму, що містить множину точок евклідового простору, пошук мінімальної відстані між еліпсоїдами, побудова гіперплощини, що розділяє два еліпсоїда. Запропоновані ефективні алгоритми для розв’язку цього класу задач.The paper is devoted to the statement and the solution of a class of quadratic optimizing problems in the computer geometry: the search of the minimum volume ellipsoid that contains the set of points of Euclidean space, the search of the minimum distance between ellipsoids, the construction of the hyperplane separating two ellipsoids. The effective algorithms for the solution of this class of problems are offered

On the ellipses and ellipsoids separation problem: an application of quantifier elimination

RESUMEN: En este trabajo se derivan fórmulas para caracterizar cuando dos elipses o dos elipsoides se superponen, tienen un contacto tangencial externo o están separados por una recta o un plano, respectivamente. Esta cuestión aparece de forma natural en la resolución de problemas en tratamiento de imágenes, robótica, modelización, etc. Esto es debido a que es muy sencillo modelar distintos tipos de objetos mediante elipses y elipsoides. Sean A y B dos elipses o dos elipsoides definidas respectivamente por medio de sus matrices asociadas XTAX = 0 y XTBX = 0. En este trabajo se analizan los resultados en \W. Wang, J. Wang, M.-S. Kim: An algebraic condition for the separation of two ellipsoids. Computer Aided Geometric Design 18, 531-539, 2001"que caracterizan la posición relativa de dos elipsoides en funcion del signo de las raíces reales del polinomio f(λ) = det(λA + B), se extienden al caso de elipses y, usando las técnicas de eliminación de cuantificadores presentadas en \J. Caravantes, L. Gonzalez-Vega: On the Interference Problem for Ellipsoids: Experiments and Applications, Lecture Notes in Computer Science 10931, 89-97, 2018", se caracteriza dicha posición relativa en función de las entradas de las matrices A y B. De esta forma se determina la posición relativa de A y B sin tener que calcular los puntos de intersección entre ambos objetos y solo es necesario evaluar las fórmulas antes mencionadas. Este trabajo incluye también una experimentación en Maple donde se muestra como funcionan las fórmulas obtenidas para determinar cuando dos elipses o dos elipsoides, en movimiento, están separados por una recta o por un plano, respectivamente.ABSTRACT: In this work, several formulas are derived to characterise when two ellipses or two ellipsoids overlap, have an external tangential contact or are separated by a line or a plane, respectively. This problem appears naturally in the resolution of problems in image processing, robotics, modeling, etc. because it is very easy to model different types of objects by using ellipses and ellipsoids. Let A and B two ellipses or two ellipsoids defined respectively by means of their associated matrices XTAX = 0 and XTBX = 0. In this work we analyse the results in \W. Wang, J. Wang, M.-S. Kim: An algebraic condition for the separation of two ellipsoids. Computer Aided Geometric Design 18, 531-539, 2001"that characterise the relative position of two ellipsoids in terms of the sign of the real roots of the polynomial f(λ) = det(λA + B), they are extended to the case of ellipses and, using the quantifier elimination techniques presented in \ J. Caravantes, L. Gonzalez-Vega: On the Interference Problem for Ellipsoids: Experiments and Applications, Lecture Notes in Computer Science 10931, 89-97, 2018", the relative position is characterised in terms of the entries of A and B. In this way, the relative position of A and B is determined without computing the points of intersection between both objects and it is only necessary to evaluate the aforementioned formulas. This work also includes an experimentation in Maple showing how the obtained formulas can be used to determine when two moving ellipses or two moving ellipsoids are separated by a line or a plane, respectively.Grado en Matemática

Equations, inequations and inequalities characterizing the configurations of two real projective conics

Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well--adapted to the study of the relative position of two conics defined by equations depending on parameters.Comment: 31 pages. See also http://emmanuel.jean.briand.free.fr/publications/twoconics/ Added references to important prior work on the subject. The title changed accordingly. Some typos and imprecisions corrected. To be published in Applicable Algebra in Engineering, Communication and Computin

Numerical modelling of ellipsoidal inclusions

Within the framework of numerical algorithms for the threedimensional random packing of granular materials this work presents an innovative formulation for polydispersed ellipsoidal particles, including an overlapping detection algorithm for an optimized simulation of the mesostructure of geomaterials, particularly concrete. Granular composite cement-based materials can be so reconstructed with adequate precision in terms of grain size distribution. Specifically, the algorithm performance towards the assumed inclusion shape (ellipsoidal or spheric) and degree of regularity (round or irregular) is here discussed. Examples on real grading curves prove that this approach is effective. The advantages of the proposed method for computational mechanics purposes are also disclosed when properly interfaced with visualization CAD (Computer Aided Design) tools

Revisión de literatura de jerarquía volúmenes acotantes enfocados en detección de colisiones

(Eng) A bounding volume is a common method to simplify object representation by using the composition of geometrical shapes that enclose the object; it encapsulates complex objects by means of simple volumes and it is widely useful in collision detection applications and ray tracing for rendering algorithms. They are popular in computer graphics and computational geometry. Most popular bounding volumes are spheres, Oriented-Bounding Boxe s (OBB’ s), Axis-Align ed Bound ing Boxes (AABB’ s); moreover , the literature review includes ellipsoids, cylinders, sphere packing, sphere shells , k-DOP’ s, convex hulls, cloud of points, and minimal bounding boxe s, among others. A Bounding Volume Hierarchy is ussualy a tree in which the complete object is represented thigter fitting every level of the hierarchy. Additionally, each bounding volume has a cost associated to construction, update, and interference te ts. For instance, spheres are invariant to rotation and translations, then they do not require being updated ; their constructions and interference tests are more straightforward then OBB’ s; however, their tightness is lower than other bounding volumes. Finally , three comparisons between two polyhedra; seven different algorithms were used, of which five are public libraries for collision detection.(Spa) Un volumen acotante es un método común para simplificar la representación de los objetos por medio de composición de formas geométricas que encierran el objeto; estos encapsulan objetos complejos por medio de volúmenes simples y son ampliamente usados en aplicaciones de detección de colisiones y trazador de rayos para algoritmos de renderización. Los volúmenes acotantes son populares en computación gráfica y en geometría computacional; los más populares son las esferas, las cajas acotantes orientadas (OBB’s) y las cajas acotantes alineadas a los ejes (AABB’s); no obstante, la literatura incluye elipses, cilindros empaquetamiento de esferas, conchas de esferas, k-DOP’s, convex hulls, nubes de puntos y cajas acotantes mínimas, entre otras. Una jerarquía de volúmenes acotantes es usualmente un árbol, en el cual la representación de los objetos es más ajustada en cada uno de los niveles de la jerarquía. Adicionalmente, cada volumen acotante tiene asociado costos de construcción, actualización, pruebas de interferencia. Por ejemplo, las esferas so invariantes a rotación y translación, por lo tanto no requieren ser actualizadas en comparación con los AABB no son invariantes a la rotación. Por otro lado la construcción y las pruebas de solapamiento de las esferas son más simples que los OBB’s; sin embargo, el ajuste de las esferas es menor que otros volúmenes acotantes. Finalmente, se comparan dos poliedros con siete algoritmos diferentes de los cuales cinco son librerías públicas para detección de colisiones

3D network modelling of fracture processes in fibre-reinforced geomaterials

The width of fracture process zones in geomaterials is commonly assumed to depend on the type of heterogeneity of the material. Still, very few techniques exist, which link the type of heterogeneity to the width of the fracture process zone. Here, fracture processes in geomaterials are numerically investigated with structural network approaches, whereby the heterogeneity in the form of large aggregates and low volume fibres is modelled geometrically as poly-dispersed ellipsoids and mono-dispersed line segments, respectively. The influence of aggregates, fibres and combinations of both on fracture processes in direct tensile tests of periodic cells is investigated. For all studied heterogeneities, the fracture process zone localises at the start of the softening regime into a rough fracture. For aggregates, the width of the fracture process zone is greater than for analyses without aggregates. Fibres also increase the initial width of the fracture process zone and, in addition, result in a widening of this zone due to fibre pull out

On the implicit equation of conics and quadrics offsets

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in ?A Treatise on the Analytic Geometry of Three Dimensions? by G. Salmon for offsets to non-degenerate conics and central quadrics.This research was funded by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P