Applications In Computational Geometry Research Articles

First-order methods for the convex hull membership problem

The convex hull membership problem (CHMP) consists in deciding whether a certain point belongs to the convex hull of a finite set of points, a decision problem with important applications in computational geometry and in foundations of linear programming. In this study, we review, compare and analyze first-order methods for CHMP, namely, Frank-Wolfe type methods, Projected Gradient methods and a recently introduced geometric algorithm, called Triangle Algorithm (TA). We discuss the connections between this algorithm and Frank-Wolfe, showing that TA can be interpreted as an inexact Frank-Wolfe. Despite this similarity, TA is strongly based on a theorem of alternatives known as distance duality. By using this theorem, we propose suitable stopping criteria for CHMP to be integrated into Frank-Wolfe type and Projected Gradient, specializing these methods to the membership decision problem. Interestingly, Frank-Wolfe integrated with such stopping criteria coincides with a greedy version of the Triangle Algorithm which is, in its turn, equivalent to an algorithm due to von Neumann. We report numerical experiments on random instances of CHMP, carefully designed to cover different scenarios, that indicate which algorithm is preferable according to the geometry of the convex hull and the relative position of the query point. Concerning potential applications, we present two illustrative examples, one related to linear programming feasibility problems and another related to image classification problems.

Method to Determine the Centroid of Non-Homogeneous Polygons Based on Suspension Theory

The centroid is most often used to describe the average position of an object’s mass and has very important applications in computational geometry, applied physics, and spatial information fields, amongst others. Based on the suspension theory of physics, this paper proposes a new method to determine the centroid of a non-homogeneous polygon by the intersection of the two balance lines. By considering the inside point value and distance to the balance line, the proposed method overcomes the traditional method’s limitation of only considering the geometric coordinates of the boundary points of the polygon. The results show that the consideration of grid distance and grid value is logical and consistent with the calculation of the centroid of a non-homogeneous polygon. While using this method, a suitable value for relative parameters needs to be established according to specific application instances. The proposed method can be applied to aid in solving specific problems such as location assessment, allocation of resources, spatial optimization, and other relative uses.

Spherical harmonic shape descriptors of nodal force demands for quantifying spatial truss connection complexity

The connections of a spatial truss structure play a critical role in the safe and efficient transfer of axial forces between members. For discrete connections, they can also improve construction efficiency by acting as registration devices that lock members in precise orientations. As more geometrically complex spatial trusses are enabled by computational workflows and the demand for material-efficient spanning systems, there is a need to understand the effects of global form on the demands at the connections. For large-scale structures with irregular geometry, customizing individual nodes to meet exact member orientations and force demands may be infeasible; conversely, standardizing all connections results in oversized nodes and a compromise in registration potential. We propose a method for quantifying the complexity of spatial truss designs by the variation in nodal force demands. By representing nodal forces as a geometric object, we leverage the spherical harmonic shape descriptor, developed for applications in computational geometry, to characterize each node by a rotation and translation-invariant fixed-length vector. We define a complexity score for spatial truss design by the variance in the positions of the feature vectors in higher-dimensional space, providing an additional performance metric during early stage design exploration. We then develop a pathway towards reducing complexity by clustering nodes with respect to their feature vectors to reduce the number of unique connectors for design while minimizing the effects of mass standardization.

Orthogonality of isometries in the conformal model of the 3D space

Motivated by questions on orthogonality of isometries, we present a new construction of the conformal model of the 3D space using just elementary linear algebra. In addition to pictures that can help the readers to understand the conformal model, our approach allows to obtain matrix representation of isometries that can be useful, for example, in applications of computational geometry, including computer graphics, robotics, and molecular geometry.

Strategy and algorithms for the parallel solution of the nearest neighborhood problem in shared-memory processors

The neighborhood problem appears in many applications of computational geometry, computational mechanics, etc. In all these situations, the main requirement for a competitive implementation is performance, which can only be attained in modern hardware by exploiting parallelism. However, whereas the performance of serial algorithms is fairly predictable, that of parallel methods depends on delicate issues that have a huge impact (cache memory, cache misses, memory alignment, etc.), but are not easy to control. Even if there is not a simple approach to deal with these factors in shared-memory architectures, it is quite convenient to program parallel algorithms where the data are segregated on a per-thread basis. With this objective in mind, we propose a strategy to develop parallel algorithms based on a two-level design, and apply it to efficiently solve the nearest neighborhood problem. At a higher level, the proposed methods orchestrate the parallel algorithm and split the space into cells stored in a hash table; at the lower level, our methods hold serial search algorithms that are completely agnostic to the high-level counterpart. Using this strategy, we have developed a library combining different serial and parallel algorithms, optimized them, and assessed their performance. The analysis carried out allows to better understand the main bottlenecks in the algorithmic solution of the nearest neighborhood problem and come out with very fast implementations that improve existing available software.

Computational Geometry Applications

Computational geometry is an integral part of mathematics and computer science deals with the algorithmic solution of geometry problems. From the beginning to today, computer geometry links different areas of science and techniques, such as the theory of algorithms, combinatorial and Euclidean geometry, but including data structures and optimization. Today, computational geometry has a great deal of application in computer graphics, geometric modeling, computer vision, and geodesic path, motion planning and parallel computing. The complex calculations and theories in the field of geometry are long time studied and developed, but from the aspect of application in modern information technologies they still are in the beginning. In this research is given the applications of computational geometry in polygon triangulation, manufacturing of objects with molds, point location, and robot motion planning.

Time complexity of two disjoint simple paths

Finding two disjoint simple paths on two given sets of points is a geometric problem introduced by Jeff Erickson. This problem has various applications in computational geometry, like robot motion planning, generating polygon etc. We will present a reduction from planar Hamiltonian path to this problem, and prove that it is NP-Complete. To the best of our knowledge, no study has considered its complexity up until now. We also present a reduction from planar Hamiltonian path problem to the problem of finding a path on given points in the presence of arbitrary obstacles and prove that it is NP-Complete too. Also, we present a heuristic algorithm with time complexity of O(n4) to solve this problem. The proposed algorithm first calculates the convex hull for each of the entry points and then produces two simple paths on the two entry point sets

Two-dimensional range successor in optimal time and almost linear space

In this article, we revisit the problem of supporting two-dimensional range successor queries. We present a data structure with O(nlg⁡lg⁡n) words of space and O(lg⁡lg⁡n) query time. This improves the work of Nekrich and Navarro (2012) by a factor of lg⁡lg⁡n in query time, or a factor of lgϵ⁡n in space cost, where ϵ is an arbitrary positive constant. Our data structure matches the state of the art for two-dimensional range emptiness queries and achieves the optimal query time. This structure has fruitful applications in computational geometry and text indexing.

A linear time algorithm for minimum conic link path in a simple polygon

We consider the problem of finding, in a simple polygon, from a starting point to a destination point, a piecewise path consisting of conic sections. By considering only one type of conic section, i.e., circular, elliptic, hyperbolic, or parabolic curves, we present an O(n) time algorithm for computing the path with the minimum number of conic sections. The studied problem is the generalization of the straight line link path version. The results can be conducted in versatile applications: the hidden surface removal problem in CAD/CAM, the contour tracing, the red-blue intersection problem, the robot motion planning, and related computational geometry applications. The linear time property is most vital for those applications need to take instant reaction.

Group Reverse kNN Query optimisation

Spatial queries have been used to assist human mobility. One of them is the Reverse Nearest Neighbour (RNN) Queries, which its concept is to have a set of objects that considers a query object as the nearest neighbour. One solution to the RNN queries is through region approach, which in this case every single object in the space has a certain region where all objects inside this region will think of the query object as their nearest neighbour. This approach is very efficient, especially when it is implemented in mobile environment. However, current RNN queries with region approach can only be applicable to a single query object. In real life, there are some cases where we want to find a region for several objects altogether, not only for a single object. Hence, we propose a solution to this problem through the Group Reverse kNN. We will find a specific region based on multiple query objects, where any objects located inside this region will always consider all of the query objects as the nearest compared to the non-query objects. The development of the algorithm is achieved through the application of computational geometry. Our experiments demonstrate the performance efficiency and accuracy of the proposed algorithms.

Effective leaf area index retrieving from terrestrial point cloud data: coupling computational geometry application and Gaussian mixture model clustering

Abstract. Leaf area index (LAI) is one of the most important structural parameters of forestry studies which manifests the ability of the green vegetation interacted with the solar illumination. Classic understanding about LAI is to consider the green canopy as integration of horizontal leaf layers. Since multi-angle remote sensing technique developed, LAI obliged to be deliberated according to the observation geometry. Effective LAI could formulate the leaf-light interaction virtually and precisely. To retrieve the LAI/effective LAI from remotely sensed data therefore becomes a challenge during the past decades. Laser scanning technique can provide accurate surface echoed coordinates with densely scanned intervals. To utilize the density based statistical algorithm for analyzing the voluminous amount of the 3-D points data is one of the subjects of the laser scanning applications. Computational geometry also provides some mature applications for point cloud data (PCD) processing and analysing. In this paper, authors investigated the feasibility of a new application for retrieving the effective LAI of an isolated broad leaf tree. Simplified curvature was calculated for each point in order to remove those non-photosynthetic tissues. Then PCD were discretized into voxel, and clustered by using Gaussian mixture model. Subsequently the area of each cluster was calculated by employing the computational geometry applications. In order to validate our application, we chose an indoor plant to estimate the leaf area, the correlation coefficient between calculation and measurement was 98.28 %. We finally calculated the effective LAI of the tree with 6 × 6 assumed observation directions.

A Fault-Tolerant Offset Algorithm for Measured Data with Defects

Offsetting of measured data, as a basic geometric operation, has already been widely used in many areas, like reverse engineering, rapid prototyping and NC machining. However, measured data always carry typical defects like caves and singular points. A fault-tolerant offset method is proposed to create the high quality offset surface of measured data with such defects. Firstly, we generated an expansion sphere model of measured data with the radius equivalent to the offset length. Secondly, using the computational geometry application of convex hull, we acquire the data of outermost enveloping surface of this expansion sphere model. Finally, we use local MLS projection fitting method to wipe out existing defects, and generate the high-quality triangular mesh surface of the offset model. The offset surface generated by this method is suitable for practical engineering application due to its high efficiency and accuracy.

Computing the Crosscap Number of a Knot Using Integer Programming and Normal Surfaces

The crosscap number of a knot is an invariant describing the nonorientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an algorithm based on exact integer programming, both of which either compute the solution precisely or reduce it to two possible values, and (iii) a fast but limited precision integer programming algorithm that bounds the solution from above.The first two algorithms advance the theoretical state-of-the-art, but remain intractable for practical use. The third algorithm is fast and effective, which we show in a practical setting by making significant improvements to the current knowledge of crosscap numbers in knot tables. Our integer programming framework is general, with the potential for further applications in computational geometry and topology.

Computing shortest homotopic cycles on polyhedral surfaces with hyperbolic uniformization metric

The problem of computing shortest homotopic cycles on a surface has various applications in computational geometry and graphics. In general, shortest homotopic cycles are not unique, and local shortening algorithms can become stuck in local minima. For surfaces with a negative Euler characteristic that can be given a hyperbolic uniformization metric, however, we show that they are unique and can be found by a simple locally shortening algorithm. We also demonstrate two applications: constructing extremal quasiconformal mappings between surfaces with the same topology, which minimize angular distortion, and detecting homotopy between two paths or cycles on a surface.

Computational Geometry for Continuous Resource Location Model in a Competitive Environment

location has become a fundamental question to be answered in the day today life. It has found its place in the life for over hundred years from the smallest application like where to locate any electric appliance in a kitchen to opening a new departmental store by a company. Depending upon the context in which the location model is being used, various objective functions may be designed while main features will always remain same (1). Here in this paper, we have studied the continuous resource location model where competing resources are already known to exist in the system under study. The newly located resources compete for its share of customer demands with the pre-existing resources in the system and with each other also (2). The paper also discusses how the applications of computational geometry relate to this spatial problem.

Applications of computational geometry to the molecular simulation of interfaces

The identification of the interfacial molecules in fluid-fluid equilibrium is a long-standing problem in the area of simulation. We here propose an alternative point of view, making use of concepts taken from the field of computational geometry, where the definition of the "shape" of a set of points is a well-known problem. In particular, we employ the alpha -shape construction which, applied to the positions of the molecules, selects a shape and identifies its boundary points, which we will take to define our interfacial molecules. A single parameter needs to be fixed (the "alpha" of the alpha shape), and several proposals are examined, all leading to very similar choices. Results of this methodology are evaluated against previous proposals, and seen to be reasonable.

A generic and flexible framework for the geometrical and topological analysis of (algebraic) surfaces

We present a generic framework on a set of surfaces S in R 3 that provides their geometric and topological analysis in order to support various algorithms and applications in computational geometry. Our implementation follows the generic programming paradigm, that is, to support a certain family of surfaces, we require a small set of types and some basic operations on them, all collected in a model of the newly presented SurfaceTraits_3 concept. The framework obtains geometric and topological information on a non-empty set of surfaces in two steps. First, important 0- and 1-dimensional features are projected onto the xy-plane, obtaining an arrangement A S with certain properties. Second, for each of its components, a sample point is lifted back to R 3 while detecting intersections with the given surfaces. For the projection we rely on Cgal's Arrangement_2 package as basic tool. Anyhow, the complexity of the output is high, and thus, we particularly regard the framework as key ingredient for querying information on and constructing geometric objects from a small set of surfaces. Examples are meshing of single surfaces, the computation of space-curves defined by two surfaces, to compute lower envelopes of surfaces, or as a basic step to compute an efficient representation of a three-dimensional arrangement. We show that the well-known family of (semi-)algebraic surfaces fulfills the framework's requirements. As robust implementations on these surfaces are lacking these days, we consider the framework to be an important step to fill this gap. In particular, we instantiate the framework by a fully-fledged model for special algebraic surfaces, namely quadrics. This instantiation already supports main tasks demanded from rotational robot motion planning, for example, as expected to compute a Piano Mover's instance.

Computational Geometry in Navigation and Path Planning [From The Guest Editors

The six contributions to this special issue focus on computational geometry in navigation and path planning. The issue presents the applications of computational geometry in obstacle avoidance, path planning with minimum clearance, autonomous mobile robot navigation, exploration of unknown polygonal environments, and it explores the practical relevance to the geographical information systems, mobile networks, risk avoidance, security, games, and online system design.

Evaluation of dosimetric margins in prostate IMRT treatment plans

This work introduces a new concept--the dosimetric margin distribution (DMD)--and uses it to explain the sensitivity of a group of prostate IMRT treatment plans to patient setup errors. Prior work simulated the effect of setup errors on 27 prostate IMRT treatment plans and found the plans could tolerate larger setup errors than predicted by the van Herk margin formula. The conjectured reason for this disagreement was a breakdown in van Herk's assumption that the planned dose distribution conforms perfectly to target structures. To resolve the disagreement, this work employed the same 27 plans to evaluate the actual margin distributions that exist between: (i) the clinical target volume (CTV) and planning target volume (PTV) and (ii) the CTV and PTV minimum dose isodose surface. These distributions were evaluated for both prostate and nodal targets. Distribution (ii) is the DMD. The dosimetric margin in a given direction determines the probability that the CTV will be underdosed due to setup errors in that direction. Averaging over 4 pi sr gives the overall probability of CTV coverage. Minimum doses for prostate and nodal PTVs were obtained from dose volume histograms. Corresponding isodose surfaces were created and converted to regions of interest (ROIs). CTV, PTV, and isodose ROIs were saved as mesh files and then imported into a computational geometry application which calculated distances between meshes (i.e., margins) in 614 discrete directions covering 4 pi sr in 10 deg increments. Measured prostate CTV-to-PTV margins were close to the nominal value of 0.5 cm specified in the treatment planning protocol. However, depending on direction, prostate dosimetric margins ranged from 0.5 to 3 cm, reflecting the imperfect conformance of the planned dose distribution to the prostate PTV. For the nodal CTV, the nominal CTV-to-PTV margin employed in treatment planning was again 0.5 cm. However, due to the planning protocol, the nodal PTV follows the surface of the nodal CTV in several places, ensuring that there is no room for rigid body motion of the nodal CTV inside the nodal PTV. Measured nodal CTV-to-PTV margins were therefore zero, while nodal dosimetric margins ranged from 0.2 to 2.8 cm. Prostate and nodal target coverage were found to be well correlated with the measured DMDs, thereby resolving the apparent disagreement with our prior results. The principal conclusion is that target coverage in the presence of setup errors should be evaluated using the DMD, rather than the CTV-to-PTV margin distribution. The DMD is a useful planning metric, which generalizes the ICRU conformity index. DMDs could vary with number of beams, beam arrangements, TPS, and treatment site.

Application of computational geometry to multiuser detection in CDMA

The maximum-likelihood multiuser detection problem in code-division multiple-access is known to be an optimization problem with an objective function that is required to be optimized over a combinatorial decision region. Conventional suboptimal detectors relax the combinatorial decision region by a convex region, without altering the objective function to be optimized. We take an approach wherein the objective function is reduced to a form appropriate for the application of a polynomial complexity algorithm in computational geometry, while keeping the decision region combinatorial. The resulting detector allows a tradeoff between performance and computational complexity. The bit-error rate performance of the detector has been found to be better than the decorrelator and the linear minimum mean-square error detectors, for the same level of complexity.