A simple linear algorithm for intersecting convex polygons

NEW RESULTS IN COMPUTATIONAL GEOMETRY RELEVANT TO PATTERN RECOGNITION IN PRACTICE

We consider problems of geometric exploration and self-deployment for simple robots that can only sense the combinatorial (non-metric) features of their surroundings. Even with such a limited sensing, we show that robots can achieve complex geometric reasoning and perform many non-trivial tasks. Specifically, we show that one robot equipped with a single pebble can decide whether the workspace environment is a simply-connected polygon and, if not, it can also count the number of holes in the environment. Highlighting the subtleties of our sensing model, we show that a robot can decide whether the environment is a convex polygon, yet it cannot resolve whether a particular vertex is convex. Finally, we show that using such local and minimal sensing, a robot can compute a proper triangulation of a polygon, and that the triangulation algorithm can be implemented collaboratively by a group of m such robots, each with Θ(n/m) memory. As a corollary of the triangulation algorithm, we derive a distributed analog of the well-known Art Gallery Theorem: a group of ⌊n/3⌋ (bounded memory) robots in our minimal sensing model can self-deploy to achieve visibility coverage of an n-vertex art gallery (polygon). This resolves an open question raised recently by Ganguli et al.

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n 2) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).

We are constructing a workbench for computational geometry. This is intended to provide a framework for the implementation, testing, demonstration and application of algorithms in computational geometry. The workbench is being written in Smalltalk/V using an Apple Macintosh II.The object-oriented model used in Smalltalk is well-suited to algorithms manipulating geometric objects. In addition, the programming environment can be easily extended, and provides excellent graphics facilities, data abstraction, encapsulation, and incremental modification.We have completed the design and implementation of the workbench platform, insofar as such a system can ever be considered complete. Among the features of the system are:an interactive graphical environment, including operations for creation and editing of geometric figures, and for the operation of algorithm on these figuresthe system supports:high-level representation-independent geometric objects (points, lines, polygons,…)geometric data structures (segment trees, range trees,…)non-geometric data structures (finger trees, splay trees, heaps, …)“standard” algorithmic tools in as general a form as possible. Algorithms currently available in the system include Tarjan and van Wyk's triangulation of a simple polygon, Fortune's Voronoi diagram,Preparata's chain decomposition, and Melkman's convex hull algorithm. tools for the animation of geometric algorithmshigh-level graphical and symbolic debugging facilitiesportability, due to the separation of the machine-independent code and the machine-dependent user-interface.automatic handling of basic operations (device-independent graphics, storage management) allowing the implementor to focus on algorithmic issuesOur group is currently working on extensions in two directions:implementing additional algorithms from two-dimensional computational geometryproviding the framework for implementations of three-dimensional algorithmsWe are also conducting comparison studies of different algorithms and data structures, including a comparison of different triangulation and convex hull algorithms for large input sizes and an empirical test of the dynamic optimality conjecture of Sleator and Tarjan using both Splay and Finger trees in the Tarjan and van Wyk triangulation.The workbench is being demonstrated during this symposium.

Counting thin and bushy triangulations of convex polygons

In this paper an algorithm for the convex polygon triangulation based on the reverse Polish notation is proposed. The formal grammar method is used as the starting point in the investigation. This idea is "translated" to the arithmetic expression field enabling application of the reverse Polish notation method. .

Geometric classification of triangulations and their enumeration in a convex polygon

Recently the authors have shown that the versatility of the reconfigurable mesh can be exploited to devise 0(1) time algorithms for a number of important computational tasks relevant to image processing, computer graphics, and computer vision. Specifically, we have shown that if one or two n-vertex (convex) polygons are pretiled, one vertex per processor, onto a reconfigurable mesh of size sqrt n X sqrt n, then a number of geometric problems can be solved in 0(1) time. These include testing an arbitrary polygon for convexity, the point location problem, the supporting lines problem, the stabbing problem, constructing the common tangents of two separable convex polygons, deciding whether two convex polygons intersect, and computing the smallest distance between the boundaries of two convex polygons. The novelty of these algorithms is that the problems are solved in the dense case. The purpose of this paper is to add to the list of problems that can be solved in 0(1) time in the dense case. The problems that we address are: determining the minimum area corner triangle for a convex polygon, determining the k-maximal vertices of a restricted class of convex polygons, updating the convex hull of a convex polygon in the presence of a set of query points, and determining a point that belongs to exactly one of two given convex polygons.

We present the first quadratic-time algorithm for the greedy triangulation of a finite planar point set, and the first linear-time algorithm for the greedy triangulation of a convex polygon.

We present the first quadratic-time algorithm for the greedy triangulation of a finite planar point set, and the first linear-time algorithm for the greedy triangulation of a convex polygon.

A generalization of diagonal flips in a convex polygon

Flipping edge-labelled triangulations

This work is all about an efficient new algorithm to find out Convex Hull for a large dataset which may or may not have duplicate values. This algorithm is basically an improvement over Graham Scan Convex Hull algorithm with the help of SSGM Sort. This algorithm is further referred as Efficient Convex Hull (ECH) algorithm throughout this article. The complexity of time taken by Efficient Convex Hull algorithm presented over here is O(n) at its best and O(n log n) in the worst scenario for a problem of size n elements. The amount of memory space taken by ECH is O(1) which breaches the lower bound rendered on it by sorting algorithms on Graham Scan convex hull. Here it would be better to understand that the type of problems convex hull deals with generally has large number of duplicate values. ECH utilizes this fact and give time complexity of small “o” that is o(n log n) which is much lesser than O(n log n). Further an application of ECH is Efficient Convex Hull Vector Network (ECHVN) is proposed by combining benefits of ECH and SVMs.

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

A linear time algorithm for max-min length triangulation of a convex polygon