16 research outputs found
The dimension of splines of arbitrary degree on a tetrahedral partition
We consider the linear space of piecewise polynomials in three variables which are globally smooth, i.e., trivariate splines. The splines are defined on a uniform tetrahedral partition , which is a natural generalization of the four-directional mesh. By using Bernstein-B{\´e}zier techniques, we establish formulae for the dimension of the splines of arbitrary degree
A Generic Lazy Evaluation Scheme for Exact Geometric Computations
We present a generic C++ design to perform efficient and exact geometric
computations using lazy evaluations. Exact geometric computations are critical
for the robustness of geometric algorithms. Their efficiency is also critical
for most applications, hence the need for delaying the exact computations at
run time until they are actually needed. Our approach is generic and extensible
in the sense that it is possible to make it a library which users can extend to
their own geometric objects or primitives. It involves techniques such as
generic functor adaptors, dynamic polymorphism, reference counting for the
management of directed acyclic graphs and exception handling for detecting
cases where exact computations are needed. It also relies on multiple precision
arithmetic as well as interval arithmetic. We apply our approach to the whole
geometric kernel of CGAL
A custom designed density estimation method for light transport
We present a new Monte Carlo method for solving the global illumination problem in environments with general geometry descriptions and light emission and scattering properties. Current Monte Carlo global illumination algorithms are based on generic density estimation techniques that do not take into account any knowledge about the nature of the data points --- light and potential particle hit points --- from which a global illumination solution is to be reconstructed. We propose a novel estimator, especially designed for solving linear integral equations such as the rendering equation. The resulting single-pass global illumination algorithm promises to combine the flexibility and robustness of bi-directional path tracing with the efficiency of algorithms such as photon mapping
Characteristics of 3D solid modeling software libraries for non-manifold modeling
The aim of this paper is to provide a review of the characteristics of 3D solid modeling software libraries – otherwise known as ’geometric modeling kernels’ in non-manifold applications. ’Non-manifold’ is a geometric topology term that means ’to allow any combination of vertices, edges, surfaces and volumes to exist in a single logical body’. In computational architectural design, the use of non-manifold topology can enhance the representation of space as it provides topological clarity, allowing architects to better design, analyze and reason about buildings. The review is performed in two parts. The review is performed in two parts. The first part includes a comparison of the topological entities’ terminology and hierarchy as used within commercial applications, kernels, and within published academic research. The second part proposes an evaluation framework to explore the kernels’ support for non-manifold topology, including their capability to represent a non-manifold structure, and in performing non-regular Boolean operations, which are suitable for non-manifold modeling
On the probability of rendezvous in graphs
In a simple graph without isolated nodes the following random experiment is carried out: each node chooses one of its neighbors uniformly at random. We say a rendezvous occurs if there are adjacent nodes and such that chooses and chooses ; the probability that this happens is denoted by . M{\'e}tivier \emph{et al.} (2000) asked whether it is true that for all -node graphs , where is the complete graph on nodes. We show that this is the case. Moreover, we show that evaluating for a given graph is a \numberP-complete problem, even if only -regular graphs are considered, for any
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
Planar Nef polyhedra and generic higher-dimensional geometry
We present two generic software projects that are part of the software library CGAL. The first part described the design of a geometry kernel for higher-dimensional Euclidian geometry and the interaction with application programs. We describe software structures, interface concepts, and their models that are based on cooordinate representation, number types, and memory layout. In the higher-dimensional software kernel the interaction between linear algebra and geometric objects and primitves is one important facet. In the actual design our users can replace number types, representation types, and the traits classes that inflate kernel functionality into our current application programs: higher-dimensional convex hulls and Delaunay tedrahedralisations. In the second part we present the realization of planar Nef polyhedra. The concept of Nef polyhedra subsumes all kinds of rectilinear polyhedral subdivisions and is therefore of general applicability within a geometric software library. The software is based on the theory of extended points and segments that allows us to reuse classical algorithmic solutions like plane sweep to realize binary operations of Nef polyhedra.Wir präsentieren zwei Softwareprojekte, die Teil der Softwarebibliothek CGAL sind. Der erste Teil beschreibt den Entwurf eines Geometriekerns für höherdimensionale euklidische Geometrie und dessen Interaktion mit Anwendungsprogrammen. Wir beschreiben die Softwarestruktur, die auf der Herausarbeitung von Schnittstellenkonzepten und ihren Modellen hinsichtlich Koordinatenrepräsentation, Zahlentypen und Speicherablage beruht. Dabei spielt im Höherdimensionalen die Interaktion zwischen linearer Algebra und den entsprechenden geometrischen Objekten und primitiven Operationen eine wesentliche Rolle. Unser Entwurf erlaubt das Auswechseln von Zahlentypen,
Repräsentations- und Traitsklassen bei der Berechnung von d-dimensionalen konvexen Hüllen und Delaunay-Simplexzerlegungen. Im zweiten Teil stellen wir die Realisierung von planaren Nef-Polyedern vor. Das Konzept der Nef-Polyeder umfasst alle linear-polyedrisch begrenzten Unterteilungen. Wir beschreiben ein Softwaremodul das umfassende Funktionalität zur Verfügung stellt. Als theoretische Grundlage des Entwurfs dient die Theorie erweiterter Punkte und Segmente, die es uns erlaubt, vorhandene Algorithmen wie z.B. Plane-Sweep zur Realisierung binärer Operationen von Nef-Polyedern zu nutzen