Revisiting variable radius circles in constructive geometric constraint solving

Variable-radius circles are common constructs in planar constraint solving and are usually not handled fully by algebraic constraint solvers. We give a complete treatment of variable-radius circles when such a circle must be determined simultaneously with placing two groups of geometric entities. The problem arises for instance in solvers using triangle decomposition to reduce the complexity of the constraint problem.Postprint (published version

Tree-decomposable and underconstrained geometric constraint problems

Preprin

A Generalized Malfatti Problem

Abstract Malfatti's problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle's sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly

Speeding up the optimal method of Drezner for the p-centre problem in the plane

This paper revisits an early but interesting optimal algorithm first proposed by Drezner to solve the continuous p-centre problem. The original algorithm is reexamined and efficient neighbourhood reductions which are mathematically supported are proposed to improve its overall computational performance. The revised algorithm yields a considerably high reduction in computational time reaching, in some cases, a decrease of 96%. This new algorithm is now able to find proven optimal solutions for large data sets with over 1300 demand points and various values of p for the first time

Direct tree decomposition of geometric constraint graphs

The evolution of constraint based geometric models is tightly tied to parametric and feature-based Computer-Aided Design (CAD) systems. Since the introduction of parametric design by Pro/Engineer in the 1980's, most major CAD systems adopted constraint based geometric models as a core technology. Constraint based geometric models allowed CAD systems to provide a more powerful data model while offering an intuitive user interface. Later on, the same models also found application to fields like linkage design, chemical modeling, computer vision and dynamic geometry. Constraint based geometric models are unevaluated models. A key problem related to constraint based geometric models is the geometric constraint based solving problem which, roughly speaking, can be stated as the problem of evaluating a constraint based model. Among the different approaches to geometric constraint solving, we are interested in graph-based Decomposition-Recombination solvers. In the graph-based constructive approach, the geometric problem is first translated into a graph whose vertices represent the set of geometric elements and whose edges are the constraints. Then the constraint problem is solved by decomposing the graph into a set of sub-problems, each sub-problem is recursively divided until reaching basic problems which are solved by a dedicated equational solver. The solution to the initial problem is computed by merging the solutions to the sub-problems. The approach used by DR-solvers has been particularly successful when the decomposition into subproblems and subsequent recombination of solutions to these subproblems can be described by a plan generated a priori, that is, a plan generated as a preprocessing step without actually solving the subsystems. The plan output by the DR-planner remains unchanged as numerical values of parameters change. Such a plan is known as a DR-plan and the unit in the solver that generates it is the DR-planner. In this setting, the DR-plan is then used to drive the actual solving process, that is, computing specific coordinates that properly place geometric objects with respect to each other. In this thesis we develop a new DR-planner algorithm for graph-constructive two dimensional DR-solvers. This DR-planner is based on the tree-decomposition of a graph. The triangle- or tree-decomposition of a graph decomposes a graph into three subgraphs such that subgraphs pairwise share one vertex. Shared vertices are called hinges. The tree-decomposition of a geometric constraint graph is in some sense the construction plan that solves the corresponding problem. The DR-planner algorithm first transforms the input graph into a simpler, planar graph. After that, an specific planar embedding is computed for the transformed graph where hinges, if any, can be straightly found. In the work we proof the soundness of the new algorithm. We also show that the worst case time performance of the resthe number of vertices of the input graph. The resulting algorithm is easy to implement and is as efficient as other known solving algorithms.L'evolució de models geomètrics basats en restriccions està fortament lligada al sistemes de Disseny Assistit per Computador (CAD) paramètrics i als basats en el paradigma de disseny per mitjà de característiques. Des de la introducció del disseny paramètric per part de Pro/Engineer en els anys 80, la major part de sistemes CAD utilitzaren com a tecnologia de base els models geomètrics basats en restriccions. Els models geomètrics basats en restriccions permeteren als sistemes CAD proporcionar un model d'informació més ampli i alhora oferir una interfície d'usuari intuitiva. Posteriorment, els mateixos models s'aplicaren en camps com el disseny de mecanismes, el modelatge químic, la visió per computador i la geometria dinàmica. Els models geomètrics basats en restriccions són models no avaluats. Un problema clau relacionat amb el models de restriccions geomètriques és el problema de la resolució de restriccions geomètriques, que es resumeix com el problema d'avaluar un model basat en restriccions. Entre els diferents enfocs de resolució de restriccions geomètriques, tractem els solvers de Descomposició-Recombinació (DR-solvers) basats en graphs. En l'enfoc constructiu basat en grafs, el problema geomètric es trasllada en un pas inicial a un graf, on els vèrtexs del graf representen el conjunt d'elements geomètrics i on les arestes corresponen a les restriccions geomètriques entre els elements. A continuació el problema de restriccions es resol descomposant el graf en un conjunt de subproblemes, cadascun dels quals es divideix recursivament fins a obtenir problemes bàsics, que sovint són operacions geomètriques realitzables, per exemple, amb regle i compàs, i que es resolen per mitjà d'un solver numèric específic. Finalment, la solució del problema inicial s'obté recombinant les solucions dels subproblemes. L'enfoc utilitzat pels DR-solvers ha esdevingut especialment interessant quan la descomposició en subproblemes i la posterior recombinació de solucions d'aquests subproblemes es pot descriure com un pla de construcció generat a priori, és a dir, un pla generat com a pas de pre-procés sense necessitat de resoldre realment els subsistemes. El pla generat pel DR-planner esdevé inalterable encara que els valors numèrics dels paràmetres canviin. Aquest pla es coneix com a DR-plan i la unitat en el solver que el genera és l'anomenat DR-planner. En aquest context, el DR-plan s'utilitza com a eina del procés de resolució en curs, és a dir, permet calcular les coordenades específiques que correctament posicionen els elements geomètrics uns respecte els altres. En aquesta tesi desenvolupem un nou algoritme que és la base del DR-planner per a DR-solvers constructius basats en grafs en l'espai bidimensional. Aquest DR-planner es basa en la descomposició en arbre d'un graf. La descomposició en triangles o arbre de descomposició d'un graf es basa en descomposar un graf en tres subgrafs tals que comparteixen un vèrtex 2 a 2. El conjunt de vèrtexs compartits s'anomenen \emph{hinges}. La descomposició en arbre d'un graf de restriccions geomètriques equival, en cert sentit, a resoldre el problema de restriccions geomètriques. L'algoritme del DR-planner en primer lloc transforma el graf proporcionat en un graf més simple i planar. A continuació, es calcula el dibuix en el pla del graf transformat, on les hinges, si n'hi ha, es calculen de manera directa. En aquest treball demostrem la correctesa del nou algoritme. Finalment, proporcionem l'estudi de la complexitat temporal de l'algoritme en cas pitjor i demostrem que és quadràtica en el nombre de vèrtexs del graf proporcionat. L'algoritme resultant és senzill d'implementar i tan eficient com altres algoritmes de resolució concret

Geometric Aspects of Holographic Bit Threads

We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions.Comment: 48 pages, multiple figures. v3: matches published versio

An Adaptive Perturbation-Based Heuristic: An Application to the Continuous p-Centre Problem

A self-adaptive heuristic that incorporates a variable level of perturbation, a novel local search and a learning mechanism is proposed to solve the p-centre problem in the continuous space. Empirical results, using several large TSP-Lib data sets, some with over 1300 customers with various values of p, show that our proposed heuristic is both effective and efficient. This perturbation metaheuristic compares favourably against the optimal method on small size instances. For larger instances the algorithm outperforms both a multi-start heuristic and a discrete-based optimal approach while performing well against a recent powerful VNS approach. This is a self-adaptive method that can easily be adopted to tackle other combinatorial/global optimisation problems. For benchmarking purposes, the medium size instances with nodes are solved optimally for the first time, though requiring a large amount of computational time. As a by-product of this research, we also report for the first time the optimal solution of the vertex p-centre problem for these TSP-Lib data sets

Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age

Simultaneous Localization and Mapping (SLAM)consists in the concurrent construction of a model of the environment (the map), and the estimation of the state of the robot moving within it. The SLAM community has made astonishing progress over the last 30 years, enabling large-scale real-world applications, and witnessing a steady transition of this technology to industry. We survey the current state of SLAM. We start by presenting what is now the de-facto standard formulation for SLAM. We then review related work, covering a broad set of topics including robustness and scalability in long-term mapping, metric and semantic representations for mapping, theoretical performance guarantees, active SLAM and exploration, and other new frontiers. This paper simultaneously serves as a position paper and tutorial to those who are users of SLAM. By looking at the published research with a critical eye, we delineate open challenges and new research issues, that still deserve careful scientific investigation. The paper also contains the authors' take on two questions that often animate discussions during robotics conferences: Do robots need SLAM? and Is SLAM solved

Physically based mechanical metaphors in architectural space planning

Physically based space planning is a means for automating the conceptual design process by applying the physics of motion to space plan elements. This methodology provides for a responsive design process, allowing a designer to easily make decisions whose consequences propagate throughout the design. It combines the speed of automated design methods with the flexibility of manual design methods, while adding a highly interactive quality and a sense of collaboration with the design. The primary assumption is that a digital design tool based on a physics paradigm can facilitate the architectural space planning process. The hypotheses are that Newtonian dynamics can be used 1) to define mechanical metaphors to represent the elements in an architectural space plan, 2) to compute architectural space planning solutions, and 3) to interact with architectural space plans. I show that space plan elements can be represented as physical masses, that design objectives can be represented using mechanical metaphors such as springs, repulsion fields, and screw clamps, that a layout solution can be computed by using these elements in a dynamical simulation, and that the user can interact with that solution by applying forces that are also models of the same mechanical objects. I present a prototype software application that successfully implements this approach. A subjective evaluation of this prototype reveals that it demonstrates a feasible process for producing space plans, and that it can potentially improve the design process because of the quality of the manipulation and the enhanced opportunities for design exploration it provides to the designer. I found that an important characteristic of this approach is that representation, computation, and interaction are all defined using the same paradigm. This contrasts with most approaches to automated space planning, where these three characteristics are usually defined in completely different ways. Also emerging from this work is a new cognitive theory of design titled 'dynamical design imagery,' which proposes that the elements in a designer's mental imagery during the act of design are dynamic in nature and act as a dynamical system, rather than as static images that are modified in a piecewise algorithmic manner

The Influence of Ca2+ Buffers on Free [Ca2+] Fluctuations and the Effective Volume of Ca2+ Microdomains

Intracellular calcium (Ca2+) plays a significant role in many cell signaling pathways, some of which are localized to spatially restricted microdomains. Ca2+ binding proteins (Ca2+ buffers) play an important role in regulating Ca2+ concentration ([Ca2+]). Buffers typically slow [Ca2+] temporal dynamics and increase the effective volume of Ca2+ domains. Because fluctuations in [Oa(2+)] decrease in proportion to the square-root of a domain\u27s physical volume, one might conjecture that buffers decrease [Ca2+] fluctuations and, consequently, mitigate the significance of small domain volume concerning Ca2+ signaling. We test this hypothesis through mathematical and computational analysis of idealized buffer-containing domains and their stochastic dynamics during free Ca2+ influx with passive exchange of both Ca2+ and buffer with bulk concentrations. We derive Langevin equations for the fluctuating dynamics of Ca2+ and buffer and use these stochastic differential equations to determine the magnitude of [Ca2+] fluctuations for different buffer parameters (e.g., dissociation constant and concentration). In marked contrast to expectations based on a naive application of the principle of effective volume as employed in deterministic models of Ca2+ signaling, we find that mobile and rapid buffers typically increase the magnitude of domain [Ca2+] fluctuations during periods of Ca2+ influx, whereas stationary (immobile) Ca2+ buffers do not. Also contrary to expectations, we find that in the absence of Ca2+ influx, buffers influence the temporal characteristics, but not the magnitude, of [Ca2+] fluctuations. We derive an analytical formula describing the influence of rapid Ca2+ buffers on [Ca2+] fluctuations and, importantly, identify the stochastic analog of (deterministic) effective domain volume. Our results demonstrate that Ca2+ buffers alter the dynamics of [Ca2+] fluctuations in a nonintuitive manner. The finding that Ca2+ buffers do not suppress intrinsic domain [Ca2+] fluctuations raises the intriguing question of whether or not [Ca2+] fluctuations are a physiologically significant aspect of local Ca2+ signaling