Point Location Algorithm Research Articles

An experimental study of point location in planar arrangements in CGAL

We study the performance in practice of various point-location algorithms implemented in CGAL (the Computational Geometry Algorithms Library), including a newly devised landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line” strategy, and a trapezoidal decomposition-based search structure. The current implementation addresses general arrangements of planar curves, including arrangements of nonlinear segments (e.g., conic arcs) and allows for degenerate input (for example, more than two curves intersecting in a single point or overlapping curves). The algorithms use exact geometric computation and thus result in the correct point location. In our landmarks algorithm (a.k.a. jump & walk), special points, “landmarks,” are chosen in a preprocessing stage, their place in the arrangement is found, and they are inserted into a data structure that enables efficient nearest-neighbor search. Given a query point, the nearest landmark is located and a “walk” strategy is applied from the landmark to the query point. We report on various experiments with arrangements composed of line segments or conic arcs. The results indicate that compared to the other algorithms tested, the landmarks approach is the most efficient, when the overall (amortized) cost of a query is taken into account, combining both preprocessing and query time. The simplicity of the algorithm enables an almost straightforward implementation and rather easy maintenance. The generic programming implementation allows versatility both in the selected type of landmarks and in the choice of the nearest-neighbor search structure. The end result is an efficient point-location algorithm that bypasses the alternative CGAL implementations in most practical aspects.

Automated generation of assembly features in layered manufacturing

PurposeThe purpose of this paper is to describe an original algorithm for point location of assembly features. Decomposing a model into several parts, for its manufacture using machine‐tools with limited dimensional capabilities, requires appropriate model reassembly. Generating and positioning assembly features are required for interlocked positioning of various parts.Design/methodology/approachAn algorithm is presented for the automated generation of assembly features over the decomposition surface.FindingsA skeleton‐based point location method is proposed and its results are discussed with respect to placements achieved on a grid.Research limitations/implicationsThe algorithm describes the placement of features on the planar decomposition surface. Future work will allow its generalization to a 3D surface.Practical implicationsA series of tests made with various parts is described, and an application of this algorithm is presented in a layered manufacturing process.Originality/valueThis paper describes an original algorithm for point location of assembly features which makes it straightforward to reassemble a decomposed part such as in layered manufacturing.

A simple entropy-based algorithm for planar point location

Given a planar polygonal subdivision S , point location involves preprocessing this subdivision into a data structure so that given any query point q , the cell of the subdivision containing q can be determined efficiently. Suppose that for each cell z in the subdivision, the probability p z that a query point lies within this cell is also given. The goal is to design the data structure to minimize the average search time. This problem has been considered before, but existing data structures are all quite complicated. It has long been known that the entropy H of the probability distribution is the dominant term in the lower bound on the average-case search time. In this article, we show that a very simple modification of a well-known randomized incremental algorithm can be applied to produce a data structure of expected linear size that can answer point-location queries in O ( H ) average time. We also present empirical evidence for the practical efficiency of this approach.

The D-tree: an index structure for planar point queries in location-based wireless services

Location-based services (LBSs), considered as a killer application in the wireless data market, provide information based on locations specified in the queries. In this paper, we examine the indexing issue for querying location-dependent data in wireless LBSs; in particular, we focus on an important class of queries, planar point queries. To address the issues of responsiveness, energy consumption, and bandwidth contention in wireless communications, an index has to minimize the search time and maintain a small storage overhead. It is shown that the traditional point-location algorithms and spatial index structures fail to achieve either objective or both. This paper proposes a new index structure, called D-tree, which indexes spatial regions based on the divisions that form the boundaries of the regions. We describe how to construct a binary D-tree index, how to process queries based on the D-tree, and how to page the binary D-tree. Moreover, two parameterized methods for partitioning the original space, called fixed grid assignment (FGA) and adaptive grid assignment (AGA), are proposed to enhance the D-tree. The performance of the D-tree is evaluated using both synthetic and real data sets. Experimental results show that the proposed D-tree outperforms the well-known indexes such as the R/sup */-tree, and that both the FGA and AGA approaches can achieve different performance trade-offs between the index search time and storage overhead by fine-tuning their algorithmic parameters.

Almost tight upper bounds for vertical decompositions in four dimensions

We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is O ( n 4+ε ), for any ε > 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O ( n 2 d −4+ε ), for any ε > 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems.

Kinetic and dynamic Delaunay tetrahedralizations in three dimensions

We describe algorithms to implement fully dynamic and kinetic three-dimensional unconstrained Delaunay triangulations, where the time evolution of the triangulation is not only governed by moving vertices but also by a changing number of vertices. We use three-dimensional simplex flip algorithms, a stochastic visibility walk algorithm for point location and in addition, we propose a new simple method of deleting vertices from an existing three-dimensional Delaunay triangulation while maintaining the Delaunay property. As an example, we analyse the performance in various cases of practical relevance. The dual Dirichlet tessellation can be used to solve differential equations on an irregular grid, to define partitions in cell tissue simulations, for collision detection etc.

A note on point location in arrangements of hyperplanes

We give an algorithm for point location in an arrangement of n hyperplanes in E d with running time poly( d,log n) and space O( n d ). The space improves on the O( n d+ ε ) bound of Meiser's algorithm [Inform. and Control 106 (1993) 286] that has a similar running time.

Planar point location for large data sets

We present an algorithm for external memory planar point location that is both effective and easy to implement. The base algorithm is an external memory variant of the bucket method by Edahiro, Kokubo and Asano that is combined with Lee and Yang's batched internal memory algorithm for planar point location. Although our algorithm is not optimal in terms of its worst-case behavior, we show its efficiency for both batched and single-shot queries by experiments with real-world data. The experiments show that the algorithm benefits from the mainly sequential disk access pattern and significantly outperforms the fastest algorithm for internal memory. Due to its simple concept, the algorithm can take advantage of multiple disks and processors in a rather straightforward yet efficient way.

A Note on Communication-Efficient Deterministic Parallel Algorithms for Planar Point Location and 2D Voronoï Diagram

In this note we describe deterministic parallel algorithms for planar point location and for building the Voronoi Diagram of n co-planar points. These algorithms are designed for BSP/CGM-like models of computation, where p processors, with local memory each, communicate through some arbitrary interconnection network. They are communication-efficient since they require, respectively, O(1) and O(log p) communication steps and local computation per step. Both algorithms require local memory.

A NOTE ON COMMUNICATION-EFFICIENT DETERMINISTIC PARALLEL ALGORITHMS FOR PLANAR POINT LOCATION AND 2D VORONOÏ DIAGRAM

In this note we describe deterministic parallel algorithms for planar point location and for building the Voronoï Diagram of n co-planar points. These algorithms are designed for BSP/CGM-like models of computation, where p processors, with [Formula: see text] local memory each, communicate through some arbitrary interconnection network. They are communication-efficient since they require, respectively, O(1) and O( log p) communication steps and [Formula: see text] local computation per step. Both algorithms require [Formula: see text] local memory.

A CASE STUDY IN ALGORITHM ENGINEERING FOR GEOMETRIC COMPUTING

The goal of this paper is to prove the applicability of algorithm engineering and software design concepts to geometric computing through a vertical case study on the implementation of planar point location algorithms. The work is presented within the framework of the GEOMLIB project, aimed at developing an easy to use, reliable, and flexible library of robust and efficient geometric algorithms. We present the criteria that have inspired the preliminary design of GEOMLIB and discuss the guidelines that we have followed in the initial implementation.

The design and implementation of panar maps in CGAL

Planar maps are fundamental structures in computational geometry. They are used to represent the subdivision of the plane into regions and have numerous applications. We describe the planar map package of CGAL--a Computational Geometry Algorithms Library. We discuss its modular design and implementation. In particular we introduce the two main classes of the design--planar maps and topological maps that enable the convenient separation between geometry and topology. The modular design is implemented using a generic programming approach. By switching a template parameter--the geometric traits class, one can use the same code for planar maps of different objects such as line segments or circular arcs. More flexibility is achieved by choosing a point location algorithm out of three implemented algorithms or plugging in an algorithm implemented by the user. The user of the planar maps package can benefit both from its flexibility and robustness. We present several examples of geometric traits classes and point location algorithms which demonstrate the possibility to adapt the general package to specific needs.

On a fundamental physical principle underlying the point location algorithm in computer graphics [for EM field theory

The issue of point location is an important problem in computer graphics and the study of efficient data structures and fast algorithms is an important research area for both computer graphics and computational geometry disciplines. When filling the interior region of a planar polygon in computer graphics, it is necessary to identify all points that lie within the interior region and those that are outside. Sutherland and Hodgman are credited for designing the first algorithm to solve the problem. Their approach utilizes vector construction and vector cross products, and forms the basis of the "odd parity" rule. To verify whether a test point is within or outside a given planar polygon, a ray from the test point is drawn extending to infinity in any direction without intersecting a vertex. If the ray intersects the polygon outline an odd number of times, the region is considered interior. Otherwise, the point is outside the region. In three-dimensional space (3-space), Lee and Preparata propose an algorithm but their approach is limited to point location relative to convex polyhedrons with vertices in 3-space. Although it is rich on optimal data structures to reduce the storage requirement and efficient algorithms for fast execution, a proof of correctness of the algorithm, applied to the general problem of point location relative to any arbitrary surface in 3-space, is absent in the literature. This paper argues that the electromagnetic field theory and Gauss's Law constitute a fundamental basis for the "odd parity" rule and shows that the "odd parity" rule may be correctly extended to point location relative to any arbitrary closed surface in 3-space.

AN ALGORITHM FOR RELOCATION OF DISTRIBUTION TRANSFORMERS

Relocation of distribution transformers is a widely used strategy for reducing the energy losses in radial distribution systems. Existing algorithms for optimal feed point location are either based on “rule of thumb�? or brute force search methods. In this paper two algorithms are presented. The first algorithm is meant for finding the location corresponding to minimum peak power loss in the system, and the other algorithm is intended to locate the feed point corresponding to minimum voltage regulation in the system. The optimal location is found using these two algorithms. The proposed algorithm is extremely simple and involves very little computation. The practicality of the proposed approach has been demonstrated by using them for determining the optimal feed points for real life distribution networks.

A fast algorithm for point-location in a finite element mesh

An algorithm for the point-location problem in 2D finite element meshes as a special case of plane straight-line graphs (PSLG) is presented. The element containing a given point P is determined combining a quadtree data structure to generate a quaternary search tree and a local search wave using adjacency information. The preprocessing construction of the search tree has a complexity ofO(n·log(n)) and requires only pointer swap operations. The query time to locate a start element for local search isO(log(n)) and the final point search by ‘point-in-polygon’ tests is independent of the total number of elements in the mesh and thus determined in constant time. Although the theoretical efficiency estimates are only given for quasi-uniform meshes, it is shown in numerical examples, that the algorithm performs equally well for meshes with extreme local refenement.

On coordinate transformations in steepest descent path and stationary point locations

AbstractTwo aspects of the problems of calculating steepest descent paths and locating stationary points on surfaces E(X), which are sources of some confusion in the literature, are addressed. These include writing proper expressions for the gradient and Hessian, and their transformation properties relative to coordinate transformations, based on the invariance of the surface E(X). The appropriate transformation is derived, based on a constrained energy minimization condition, to achieve what we call the Hessian eigenvalue representation. This not only allows decoupling of the variables, but also points to the minimization direction and preserves the eigenvalues of the Hessian. These results allow one to use the steepest descent path and stationary point location algorithms in any coordinate system and obtain invariant results. The validity of these considerations are also confirmed through numerical examples. The stationary condition with constrained kinematic path length is also shown to yield a Hessian eigenvalue representation for the normal modes for small vibrations. Lastly, we have constructed a mathematically consistent definition of mass‐weighted Cartesians where the intrinsic reaction path of Fukui is a steepest descent path. © 1992 John Wiley & Sons, Inc.

A fast planar partition algorithm, II

Randomized, optimal algorithms to find a partition of the plane induced by a set of algebraic segments of a bounded degree, and a set of linear chains of a bounded degree, are given. This paper also provides a new technique for clipping, called virtual clipping , whose overhead per window W depends logarithmically on the number if intersections between the borders of W and the input segments. In contrast, the overhead of the conventional clipping technique depends linearly on this number of intersections. As an application of virtual clipping, a new simple and efficient algorithm for plannar point location is given.

Hidden surface removal for rectangles

A simple but important special case of the hidden surface removal problem is one in which the scene consists of n rectangles with sides parallel to the x- and y-axes, with viewpoint at z=∞ (that is, an orthographic projection). This special case has application to overlapping windows in computer displays. An algorithm with running time O( n log n + k log n) is given for static scenes, where k is the number of line segments in the output. Algorithms are given for a dynamic setting (that is, rectangles may be inserted and deleted) that take time O(log 2 n log log n + k log 2 n) per insert or delete, where k is now the number of visible line segments that change (appear or disappear). Algorithms for point location in the visible scene are also given.

An algorithm for generalized point location and its applications

We show that Collins' classical quantifier elimination procedure contains most of the ingredients for an efficient point location algorithm in higher-dimensional space. This leads to a polynomial-size data structure that allows us to locate, in logarithmic time, a point among a collection of real algebraic varieties of constant maximum degree, assuming that the dimension of the ambient space is fixed. This result has theoretical bearings on a number of optimization problems posed in the literature. It also gives a method for solving multidimensional searching problems in polynomial space and logarithmic query time.

Planar Point Location Using Persistent Search Treesa

A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rather complicated. We propose a simple O(log n)-query-time, O(n)-space solution, using persistent search trees. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed. We develop a persistent form of binary search tree that supports insertions and deletions in the present and queries in the past. The time per query or update is O(log m), where m is the total number of updates, and the space needed is O(1) per update. Our planar point location algorithm is an immediate application of this data structure. The structure also provides an alternative to Chazelle's hive graph structure, which has a variety of applications in geometric retrieval.