Method of orienting curves for determining the convex hull of a finite set of points in the plane

Optimization ◽  
2010 ◽  
Vol 59 (2) ◽  
pp. 175-179 ◽  
Author(s):  
Phan Thanh An
Keyword(s):  
Convex Hull ◽  
Finite Set ◽  
Set Of Points

Comments on convex hull of a finite set of points in two dimensions

1979 ◽  
Vol 9 (3) ◽  
pp. 141-142 ◽  
Author(s):  
Ferenc Dévai ◽  
Tibor Szendrényi
Keyword(s):  
Convex Hull ◽  
Two Dimensions ◽  
Finite Set ◽  
Set Of Points

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 A Finite Calculus Approach to Ehrhart Polynomials

10.37236/340 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Steven V. Sam ◽  
Kevin M. Woods
Keyword(s):  
Convex Hull ◽  
Discrete Version ◽  
Ehrhart Polynomials ◽  
Finite Set ◽  
The Individual ◽  
Set Of Points ◽  
Rational Polytope

A rational polytope is the convex hull of a finite set of points in ${\Bbb R}^d$ with rational coordinates. Given a rational polytope ${\cal P} \subseteq {\Bbb R}^d$, Ehrhart proved that, for $t\in{\Bbb Z}_{\ge 0}$, the function $\#(t{\cal P} \cap {\Bbb Z}^d)$ agrees with a quasi-polynomial $L_{\cal P}(t)$, called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart–Macdonald theorem on reciprocity.

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).

scholarly journals ON THE BOUNDEDNESS OF AN ITERATION INVOLVING POINTS ON THE HYPERSPHERE

2012 ◽  
Vol 22 (06) ◽  
pp. 499-515
Author(s):  
THOMAS BINDER ◽  
THOMAS MARTINETZ
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Upper Bounds ◽  
Maximal Length ◽  
Unit Hypersphere ◽  
Finite Set ◽  
Set Of Points

For a finite set of points X on the unit hypersphere in ℝd we consider the iteration ui+1 = ui + χi, where χi is the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of ui independently of X. Precisely, this upper bound is infinity for d ≥ 3 and [Formula: see text] for d = 2.

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