75 research outputs found
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
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Analysis and Manipulation of Repetitive Structures of Varying Shape
Self-similarity and repetitions are ubiquitous in man-made and natural objects. Such structural regularities often relate to form, function, aesthetics, and design considerations. Discovering structural redundancies along with their dominant variations from 3D geometry not only allows us to better understand the underlying objects, but is also beneficial for several geometry processing tasks including compact representation, shape completion, and intuitive shape manipulation. To identify these repetitions, we present a novel detection algorithm based on analyzing a graph of surface features. We combine general feature detection schemes with a RANSAC-based randomized subgraph searching algorithm in order to reliably detect recurring patterns of locally unique structures. A subsequent segmentation step based on a simultaneous region growing is applied to verify that the actual data supports the patterns detected in the feature graphs. We introduce our graph based detection algorithm on the example of rigid repetitive structure detection. Then we extend the approach to allow more general deformations between the detected parts. We introduce subspace symmetries whereby we characterize similarity by requiring the set of repeating structures to form a low dimensional shape space. We discover these structures based on detecting linearly correlated correspondences among graphs of invariant features. The found symmetries along with the modeled variations are useful for a variety of applications including non-local and non-rigid denoising. Employing subspace symmetries for shape editing, we introduce a morphable part model for smart shape manipulation. The input geometry is converted to an assembly of deformable parts with appropriate boundary conditions. Our method uses self-similarities from a single model or corresponding parts of shape collections as training input and allows the user also to reassemble the identified parts in new configurations, thus exploiting both the discrete and continuous learned variations while ensuring appropriate boundary conditions across part boundaries. We obtain an interactive yet intuitive shape deformation framework producing realistic deformations on classes of objects that are difficult to edit using repetition-unaware deformation techniques
ESOLID—a system for exact boundary evaluation
We present a system, ESOLID, that performs exact boundary evaluation of low-degree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating point filters, arbitrary floating point arithmetic with error bounds, and lower dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRL-CAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixed-precision floating point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating point boundary evaluation system on most cases
Toric Topology
Toric topology emerged in the end of the 1990s on the borders of equivariant
topology, algebraic and symplectic geometry, combinatorics and commutative
algebra. It has quickly grown up into a very active area with many
interdisciplinary links and applications, and continues to attract experts from
different fields.
The key players in toric topology are moment-angle manifolds, a family of
manifolds with torus actions defined in combinatorial terms. Their construction
links to combinatorial geometry and algebraic geometry of toric varieties via
the related notion of a quasitoric manifold. Discovery of remarkable geometric
structures on moment-angle manifolds led to seminal connections with the
classical and modern areas of symplectic, Lagrangian and non-Kaehler complex
geometry. A related categorical construction of moment-angle complexes and
their generalisations, polyhedral products, provides a universal framework for
many fundamental constructions of homotopical topology. The study of polyhedral
products is now evolving into a separate area of homotopy theory, with strong
links to other areas of toric topology. A new perspective on torus action has
also contributed to the development of classical areas of algebraic topology,
such as complex cobordism.
The book contains lots of open problems and is addressed to experts
interested in new ideas linking all the subjects involved, as well as to
graduate students and young researchers ready to enter into a beautiful new
area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography,
index. 495 pages. Comments and suggestions are very welcom
Recommended from our members
Geometry-based structural analysis and design via discrete stress functions
This PhD thesis proposes a direct and unified method for generating global static equilibrium
for 2D and 3D reciprocal form and force diagrams based on reciprocal discrete stress
functions. This research combines and reinterprets knowledge from Maxwell’s 19th century
graphic statics, projective geometry and rigidity theory to provide an interactive design and
analysis framework through which information about designed structural performance can be
geometrically encoded in the form of the characteristics of the stress function. This method
results in novel, intuitive design and analysis freedoms.
In contrast to contemporary computational frameworks, this method is direct and analytical.
In this way, there is no need for iteration, the designer operates by default within
the equilibrium space and the mathematically elegant nature of this framework results in its
wide applicability as well as in added educational value. Moreover, it provides the designers
with the agility to start from any one of the four interlinked reciprocal objects (form diagram,
force diagram, corresponding stress functions).
This method has the potential to be applied in a wide range of case studies and fields.
Specifically, it leads to the design, analysis and load-path optimisation of tension-and compression
2D and 3D trusses, tensegrities, the exoskeletons of towers, and in conjunction
with force density, to tension-and-compression grid-shells, shells and vaults. Moreover, the
abstract nature of this method leads to wide cross-disciplinary applicability, such as 2D and
3D discrete stress fields in structural concrete and to a geometrical interpretation of yield line
theory
Boolean operations on 3D selective Nef complexes : data structure, algorithms, optimized implementation, experiments and applications
Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. Consequently, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. The generality of Nef complexes is essential for some applications. In this thesis, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysisNef-Polyeder sind d-dimensionale Punktmengen, die durch eine endliche Anzahl boolescher Operationen über Halbräumen generiert werden. Sie sind abgeschlossen hinsichtlich boolescher und topologischer Operationen. Als Konsequenz daraus können sie nicht-mannigfaltige Situationen, offene und geschlossene Mengen und gemischt-dimensionale Komplexe darstellen. Die Allgemeinheit von Nef-Komplexen ist unentbehrlich für einige Anwendungen. In dieser Doktorarbeit stellen wir eine neue Datenstruktur vor, die eine Randdarstellung von dreidimensionalen Nef-polyedern und Algorithmen für boolesche Operationen realisiert. Wir benutzen exakte Arithmetik um die bekannten Probleme mit Gleitkommaarithmetik und Degeneriertheiten zu vermeiden. Außerdem präsentieren wir wichtige Optimierungen der Algorithmen und bewerten die optimierte Implementierung an Hand umfassender Experimente. Weitere Experimente belegen die theoretische Laufzeitanalyse und vergleichen unsere Implementation mit dem kommerziellen CAD kernel ACIS. ACIS is meistens bis zu sechs mal schneller, aber es gibt auch Beispiele bei denen ACIS scheitert. Nef-Polyeder können bei einer Vielzahl von Anwendungen eingesetzt werden. Wir präsentieren einfache Implementationen zweier Anwendungen - von der visuellen Hülle und von der Minkowski-Summe zwei abgeschlossener Nef-Polyeder
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