GEODESIC DISKS AND CLUSTERING IN A SIMPLE POLYGON

2011 ◽  
Vol 21 (06) ◽  
pp. 595-608 ◽  
Author(s):  
MAGDALENE G. BORGELT ◽  
MARC VAN KREVELD ◽  
JUN LUO
Keyword(s):  
Geodesic Distance ◽  
Simple Polygon ◽  
Polygonal Path ◽  
Geodesic Distances ◽  
Set Of Points

Let P be a simple polygon of n vertices and let S be a set of N points lying in the interior of P. A geodesic diskGD(p,r) with center p and radius r is the set of points in P that have a geodesic distance ≤ r from p (where the geodesic distance is the length of the shortest polygonal path connection that lies in P). In this paper we present an output sensitive algorithm for finding all N geodesic disks centered at the points of S, for a given value of r. Our algorithm runs in [Formula: see text] time, for some constant c and output size k. It is the basis of a cluster reporting algorithm where geodesic distances are used.

FINDING AN OPTIMAL BRIDGE BETWEEN TWO POLYGONS

2002 ◽  
Vol 12 (03) ◽  
pp. 249-261 ◽  
Author(s):  
XUEHOU TAN
Keyword(s):  
Hierarchical Structure ◽  
Shortest Path ◽  
Line Segment ◽  
Euclidean Distance ◽  
Geodesic Distance ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Search Trees ◽  
Bridge Problem ◽  
Path Queries

Let π(a,b) denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a,b), denoted by gd(a,b), in contrast with the Euclidean distance between a and b in the plane, denoted by d(a,b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge) that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′,p), d(p,q) and gd(q,q′), with any p′ ∈ P and any q′ ∈ Q, is minimized. We present an O(n log 3 n) time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n2) time bound. Our result is obtained by making substantial use of a hierarchical structure that consists of segment trees, range trees and persistent search trees, and a structure that supports dynamic ray shooting and shortest path queries as well.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

scholarly journals NLP Formulation for Polygon Optimization Problems

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 24 ◽  
Author(s):  
Saeed Asaeedi ◽  
Farzad Didehvar ◽  
Ali Mohades
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problems ◽  
Theoretical Perspective ◽  
Simple Polygon ◽  
Programming Models ◽  
Minimum Area ◽  
Simple Polygons ◽  
Maximum Angle ◽  
Optimal Approach ◽  
Set Of Points

In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by α + π where α ( 0 ≤ α ≤ π ) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate these three generalized problems as nonlinear programming models, and then present a genetic algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

1995 ◽  
Vol 05 (03) ◽  
pp. 243-256 ◽  
Author(s):  
DAVID RAPPAPORT
Keyword(s):  
Convex Hull ◽  
General Problem ◽  
Line Segment ◽  
Simple Polygon ◽  
Fixed Number ◽  
Line Segments ◽  
Geometric Objects ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

scholarly journals ALGORITHMS FOR DISTANCE PROBLEMS IN PLANAR COMPLEXES OF GLOBAL NONPOSITIVE CURVATURE

2014 ◽  
Vol 24 (01) ◽  
pp. 1-38 ◽  
Author(s):  
DANIELA MAFTULEAC
Keyword(s):  
Data Structure ◽  
Convex Hull ◽  
Metric Spaces ◽  
Linear Time ◽  
Single Point ◽  
Simple Polygon ◽  
Query Point ◽  
Planar Complex ◽  
Finite Set ◽  
Set Of Points

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex [Formula: see text] with n vertices, one can construct in O(n2 log n) time a data structure [Formula: see text] of size O(n2) so that, given a point [Formula: see text], the shortest path γ(x, y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n2 log n + nk log k) time, using a data structure of size O(n2 + k).

An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria

2011 ◽  
Vol 10 (5) ◽  
pp. 1113-1131 ◽  
Author(s):  
Songting Luo ◽  
Shingyu Leung ◽  
Jianliang Qian
Keyword(s):  
Variational Problem ◽  
Traffic Congestion ◽  
Geodesic Distance ◽  
State Equation ◽  
Nonlinear Functional ◽  
Congested Traffic ◽  
Adjoint State ◽  
State Method ◽  
Geodesic Distances ◽  
Adjoint State Method

AbstractThe equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.

scholarly journals NLP Formulation for Polygon Optimization Problems

2018 ◽  
Author(s):  
Saeed Asaeedi ◽  
farzad didehvar ◽  
Ali Mohades
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problems ◽  
Theoretical Perspective ◽  
Simple Polygon ◽  
Programming Models ◽  
Minimum Area ◽  
Simple Polygons ◽  
Maximum Angle ◽  
Optimal Approach ◽  
Set Of Points

In this paper, we generalize the problems of finding simple polygons with the minimum area, maximum perimeter and maximum number of vertices so that they contain a given set of points and their angles are bounded by $\alpha+\pi$ where $\alpha$ ($0\leq\alpha\leq \pi$) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with the minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate the three generalized problems as nonlinear programming models, and then present a Genetic Algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.

AVERAGE GEODESIC DISTANCE OF SIERPINSKI GASKET AND SIERPINSKI NETWORKS

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750044 ◽  
Author(s):  
SONGJING WANG ◽  
ZHOUYU YU ◽  
LIFENG XI
Keyword(s):  
Complex Networks ◽  
Finite Type ◽  
Geodesic Distance ◽  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Geodesic Distances

The average geodesic distance is concerned with complex networks. To obtain the limit of average geodesic distances on growing Sierpinski networks, we obtain the accurate value of integral in terms of average geodesic distance and self-similar measure on the Sierpinski gasket. To provide the value of integral, we find the phenomenon of finite pattern on integral inspired by the concept of finite type on self-similar sets with overlaps.

Video Graphics and High-Voltage Electron Microscopy For Analysis of Microtubule Distributions In Fixed Cells

1990 ◽  
Vol 48 (1) ◽  
pp. 524-525
Author(s):  
Richard Mcintosh ◽  
David Mastronarde ◽  
Kent McDonald ◽  
Rubai Ding
Keyword(s):  
Electron Microscopy ◽  
Cross Sections ◽  
Cell Shape ◽  
Video Camera ◽  
Computer Memory ◽  
High Voltage Electron Microscopy ◽  
Thin Sections ◽  
Voltage Electron ◽  
Morphogenetic Event ◽  
Set Of Points

Microtubules (MTs) are cytoplasmic polymers whose dynamics have an influence on cell shape and motility. MTs influence cell behavior both through their growth and disassembly and through the binding of enzymes to their surfaces. In either case, the positions of the MTs change over time as cells grow and develop. We are working on methods to determine where MTs are at different times during either the cell cycle or a morphogenetic event, using thin and thick sections for electron microscopy and computer graphics to model MT distributions.One approach is to track MTs through serial thin sections cut transverse to the MT axis. This work uses a video camera to digitize electron micrographs of cross sections through a MT system and create image files in computer memory. These are aligned and corrected for relative distortions by using the positions of 8 - 10 MTs on adjacent sections to define a general linear transformation that will align and warp adjacent images to an optimum fit. Two hundred MT images are then used to calculate an “average MT”, and this is cross-correlated with each micrograph in the serial set to locate points likely to correspond to MT centers. This set of points is refined through a discriminate analysis that explores each cross correlogram in the neighborhood of every point with a high correlation score.

Sign in / Sign up

Export Citation Format

Share Document