Showing papers on "Computational geometry published in 2022"

Optimal Convex Hull Formation on a Grid by Asynchronous Robots With Lights

Abstract: We consider the distributed setting of $n$ autonomous mobile robots that operate in Look-Compute-Move cycles and communicate with other robots using a constant number of colored lights (the robots with lights model). We assume obstructed visibility where collinear robots do not see each other. In addition, we consider a grid-based terrain embedded in the 2-dimensional euclidean plane. The Convex Hull Formation problem is to relocate the $n$ robots (starting at arbitrary, but distinct, initial positions) so that each robot is positioned on a vertex of a convex hull. In this article, we provide a framework for solving Convex Hull Formation . We then provide four asynchronous algorithms under this framework. Key measures of the algorithms’ performance include the time taken and the space occupied. The presented algorithms are randomized and their time bounds hold with high probability. The first $O(\max \lbrace n^{2},D\rbrace)$ -time, $O({n^{2}})$ -perimeter, and $O({n^{3}})$ -area algorithm serves to introduce key ideas, where $D$ is the diameter of the initial configuration. The subsequent algorithms, differing in computational requirements, run in $O(\max \lbrace n^{\frac{3}{2}},D\rbrace)$ time with a perimeter of $O(n^{\frac{3}{2}})$ and area of $O(n^{3})$ . We also prove lower bounds of $\Omega (n^{\frac{3}{2}})$ for time and perimeter and $\Omega (n^{3})$ for area, for any Convex Hull Formation algorithm; i.e., our $O(\max \lbrace n^{\frac{3}{2}},D\rbrace)-$ time algorithm is optimal in time, perimeter, and area.

Procedural Modeling with Blender's Geometry Nodes: A workshop on taking advantage of the Geometry Nodes feature in Blender for procedural modeling

Abstract: Since about a year ago, the 3D program Blender has shipped with a set of procedural/parametric modeling tools called Geometry Nodes. While still fairly new and in development, the system has already unlocked several new ways for users to express their creativity and solve practical problems. It allows for the creation of new geometry from scratch, modification of existing geometry, and distribution of instances using a node graph. This presentation will cover those three major use cases, what's possible to create with the system, and demonstrate how to work with the tools by showing how to create a simple model generator.

Complex Shape Modelling by Tessellation. Advanced Geometry Control for Variably Curved Surfaces

Abstract: . Tessellation is a flexible modelling method for computational designers that allows to articulate and detail complex topological surfaces by repeating components along target surfaces. This technique provides a discretization strategy compatible with many digital-fabrication processes. Well-known geometrical issues limit such an approach on surfaces characterized by highly variable curvature, which can output errors such as self-intersections and undesired deformations. This research shows geometrical methods to manage such problems and implement them in an open-source tool. In particular, automatic and manual procedures for manipulating normal vector fields are utilized and applied to the modelling of complex shape case studies.

Optimal Algorithms for Geometric Centers and Depth

Abstract: We develop a general randomized technique for solving implicit linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many cases, the structure of the implicitly defined constraints can be used to obtain faster linear program solvers. We apply this technique to obtain near-optimal algorithms for a variety of fundamental problems in geometry. For a given point set $P$ of size $n$ in $\mathbb{R}^d$, we develop algorithms for computing geometric centers of a point set, including the centerpoint and the Tukey median, and several other more involved measures of centrality. For $d=2$, the new algorithms run in $O(n\log n)$ expected time, which is optimal, and for higher constant $d>2$, the expected time bound is within one logarithmic factor of $O(n^{d-1})$, which is also likely near optimal for some of the problems.

A corner smoothing approach for CNC machines based on $\eta^{3D}$-splines

Abstract: The efficiency of Computer Numerical Control (CNC) machines and the quality level of their products can be improved if GCode paths are properly smoothed through the use of junction primitives, so as to achieve a high-level of geometric continuity. In this work, a planning primitive named $\eta^{3D}$-splines is used for the generation of composite paths characterized by the third order geometric continuity. The tolerance between the original and the smoothed path can be imposed during the planning process. The strength of the proposed strategy is represented by its straightforward implementation: the $\eta^{3D}$-splines coefficients are directly and efficiently computed, through closed form expressions, from the assigned interpolating conditions. Additionally, smooth junctions can be easily created even when the original path includes circular segments. The low computational burden of the proposed strategy makes it suited for the real-time generation of smooth composite paths. Comparisons are proposed in the paper with an analogous technique based on the 3D general clothoids.

The Use of Computational Geometry Techniques to Resolve the Issues of Coverage and Connectivity in Wireless Sensor Networks

Abstract: Wireless Sensor Networks (WSNs) enhance the ability to sense and control the physical environment in various applications. The functionality of WSNs depends on various aspects like the localization of nodes, the strategies of node deployment, and a lifetime of nodes and routing techniques, etc. Coverage is an essential part of WSNs wherein the targeted area is covered by at least one node. Computational Geometry (CG) -based techniques significantly improve the coverage and connectivity of WSNs. This paper is a step towards employing some of the popular techniques in WSNs in a productive manner. Furthermore, this paper attempts to survey the existing research conducted using Computational Geometry-based methods in WSNs. In order to address coverage and connectivity issues in WSNs, the use of the Voronoi Diagram, Delaunay Triangulation, Voronoi Tessellation, and the Convex Hull have played a prominent role. Finally, the paper concludes by discussing various research challenges and proposed solutions using Computational Geometry-based techniques.

NP-ConicJS: The JavaScript Library of General Conic in Dynamic Geometry

Abstract: The application of computer and Web technology in teaching has become an inevitable trend of modern education. The Dynamic Geometry System brings innovation to traditional geometry teaching. Conic curves have always been one of the most important topics of geometric research. There is a lack of an algorithm library for dynamic geometric conic curves that are based on JavaScript. In this paper, an efficient and reusable conic curve algorithm library is proposed based on the general theory of conic curves. The algorithm library implements the functions of conic curve construction, transformation and related property calculation. Finally, the application in NetPad shows that the algorithm library brings convenience to the teaching of conic curve geometry and has important practical application value.

Geometric Bases of Parallel Computing in Computer Modeling and Computer-Aided Design Systems

Abstract: The concept of developing a geometric CAD kernel based on the invariants of parallel projection of geometric objects on the axes of the global coordinate system, which combines the potential of constructive geometric modeling methods that can provide paralleling of geometric constructions by tasks (message passing), and the mathematical apparatus "Point calculus" capable of implementing data paralleling by means of subordinate calculations (data parallel) is proposed. Use of subordinate calculation of point equations allows not only to parallelize calculations along coordinate axes, but also to provide coherence of computational operations by threads, which significantly reduces downtime and optimizes the performance of CPU to achieve the maximum effect of parallel computations. The greater the dimensionality of the modeled geometric object, the more it lends itself to paralleling computational flows. This leads to the fact that the computation time of a multidimensional problem becomes a value independent of the number of measurements. All calculations will run in parallel and finish simultaneously. The example of parallel computational algorithm for topographic surface modeling demonstrates the possibilities of realization of the offered concept for definition of continuous and discrete geometrical objects, the analytical description of which is carried out in point-calculus. As a result, to build a single 16-point patches, the distribution of parallel computing on 12 threads for the 4 direction lines and 3 threads for the formative line is obtained. Further, the number of simultaneously involved computational threads is a value proportional to the number of 16-point patches and can be further optimized by calculating several forming lines in parallel. In the above example, all computational threads are fully balanced in the number of calculations, which greatly minimizes the downtime of calculations and optimizes the performance of the processor. Also the proposed approach to the organization of parallel computations can be effectively used for the numerical solution of differential equations using geometric interpolants, which together with the development of models of geometric objects in the point calculus creates a closed loop digital production, which by analogy with the isogeometric method eliminates the need to coordinate geometric information in the interaction between CAD and FEA systems.

An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons

Abstract: Given in the plane a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P. It is known that the problem has an $$\Omega (n+m\log m)$$ time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that solves the problem in $$O(n+m\log m)$$ expected time. The previous best deterministic algorithms solve the problem in $$O(n\log \log n+ m\log m)$$ time [Oh, Barba, and Ahn, SoCG 2016] or in $$O(n+m\log m+m\log ^2\!n)$$ time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm that takes $$O(n+m\log m)$$ time, which is optimal. This answers affirmatively an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.

Application of Adjusted Differential Evolution in Optimal Sensor Placement for Interior Coverage

Abstract: It is well known that determining visual sensors in 2D space can be often modeled as an Art Gallery problem. Tasks such as surveillance dictate the coverage of the interior of a non-convex polygon with the optimal number of sensors. The optimal sensor placement is a difficult combinatorial optimization problem, and it can be formulated as seeking the smallest number of sensors obliged to cover every point in a heterogeneous setting. In this article, we propose a suboptimal deterministic algorithm, as well as an adapted differential evolution algorithm for tackling sensor placement. Both versions of novel algorithms have been implemented and tested over hundreds of random polygons. According to the outcomes presented in the experimental analysis, it can be noticed that the approach based on differential evolution beats the deterministic technique as well as other stochastic optimization algorithms for practically all instances.

Processing the overlay of geometry segments in solving hydrophysics problems by the finite difference method

Abstract: The article deals with issues related to increasing the efficiency of working with data on the geometry of the computational domain when solving hydrophysics problems using the finite difference method. The model problem is a system of equations of the pollutant distribution, including the oil and its refined products, in the computational domain – Azov Sea. To describe the computational domain, a model of a two-dimensional computational grid is used, which is used in the implementation of numerical calculations. Class diagrams are presented for describing the geometry of the object under study, as well as its constituent segments. In order to improve the performance of calculations, an algorithm for combining geometry segments was developed, in which the original algorithm was divided into separate fragments by introducing a number of conditional structures. As a result of experimental data processing, regression equations were obtained that describe the dependence of the algorithm execution time on the number of joins. The developed algorithm and class library make it possible to work with the description of the geometry of the object under study as a set of parameterized primitives and educe the time spent on the formation of the description of the computational domain by up to 27%.

Constructing L∞ Voronoi Diagrams in 2D and 3D

Abstract: Voronoi diagrams and their computation are well known in the Euclidean L2 space. They are easy to sample and render in generalized Lp spaces but nontrivial to construct geometrically. Especially the limit of this norm with p → ∞ lends itself to many quad‐ and hex‐meshing related applications as the level‐set in this space is a hypercube. Many application scenarios circumvent the actual computation of L∞ diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis‐aligned sites. Our novel algorithm allows for the construction of generalized L∞ Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level‐set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non‐cell geometry is filtered out based on the L∞ Voronoi criterion, leaving only the non‐convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L∞ diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm.