Finding the convex hull of a simple polygon

We describe a new algorithm for finding the convex hull of any simple polygon specified by a sequence of m vertices.An earlier convex hull finder of ours is limited to polygons which remain simple (i.e., nonselfintersecting) when locally non-convex vertices are removed. In this paper we amend our earlier algorithm so that it finds with complexity O(m) the convex hull of any simple polygon, while retaining much of the simplicity of the earlier algorithm.     

Time- and storage-efficient implementation of an optimal planar convex hull algorithm

An implementation of an algorithm for computing the convex hull of a finite planar set of points is presented. The program is compared with an algorithm for the same purpose coded previously. Experimental results indicate that our program is superior to the other in terms of both running time and storage requirements.     

On the ultimate convex hull algorithm in practice

Kirkpatrick and Seidel [13,14] recently proposed an algorithm for computing the convex hull of n points in the plane that runs in O(n log h) worst case time, where h denotes the number of points on the convex hull of the set. Here a modification of their algorithm is proposed that is believed to run in O(n) expected time for many reasonable distributions of points. The above O(n log h) algorithmsare experimentally compared to the O(n log n) ‘throw-away’ algorithms of Akl, Devroye and Toussaint [2, 8, 20]. The results suggest that although the O(n Log h) algorithms may be the ‘ultimate’ ones in theory, they are of little practical value from the point of view of running time.     

A counterexample to an algorithm for computing monotone hulls of simple polygons

A two-stage algorithm was recently proposed by Sklansky (1982) for computing the convex hull of a simple polygon P. The first step is intended to compute a simple polygon $\text{P}{}^1$ which is monotonic in both the x and y directions and which contains the convex hull vertices of P. The second step applies a very simple convex hull algorithm on $\text{P}{}^1$. In this note we show that the first step does not always work correctly and can even yield non-simple polygons, invalidating the use of the second step. It is also shown that the first step can discard convex hull vertices thus invalidating the use of any convex hull algorithm in the second step.     

Parallel algorithms for separation of two sets of points and recognition of digital convex polygons

Given two finite sets of points in a plane, the polygon separation problem is to construct a separating convexk-gon with smallestk. In this paper, we present a parallel algorithm for the polygon separation problem. The algorithm runs inO(logn) time on a CREW PRAM withn processors, wheren is the number of points in the two given sets. The algorithm is cost-optimal, since (n logn) is a lower-bound for the time needed by any sequential algorithm. We apply this algorithm to the problem of finding a convex polygon, with the minimal number of edges, for which a given convex region is its digital image. The algorithm in this paper constructs one such polygon with possibly two more edges than the minimal one.The research is sponsored by NSERC Operating Grant OGPIN 007.     

A new linear algorithm for triangulating monotone polygons

Let P = (p₁, p₂,…,p_n) be a monotone polygon whose vertices are specified in terms of cartesian coordinates in order. A new simple two-step procedure is presented for triangulating P, without the addition of new vertices, in O(n) time. Unlike the previous algorithm no specialized code is needed since the new approach uses well-known existing algorithms for first decomposing P into edge-visible polygons and subsequently triangulating these.     

Optimal algorithms for computing the minimum distance between two finite planar sets

It is shown in this paper that the minimum distance between two finite planar sets of n points can be computer in O(n log n) worst-case running time and that this is optimal to within a constant factor. Furthermore, when the sets form a convex polygon this complexity can be reduced O(n).     

Comments on two parallel algorithms for the planar convex hull problem

In this note some comments on the paper: D.J. Evans and Shao-wen Mai, Two parallel algorithms for the convex hull problem in a two dimensional space, Parallel Computing 2 (1985) 313–326 are given.     

On the conditions for success of sklansky''s convex hull algorithm

The convex hull algorithm for simple polygons, due to Sklansky, fails in some cases, but its extreme simplicity, compared to the later algorithms, revived an interest in this algorithm. A sufficient condition for its success was given recently by Toussaint and Avis. They have proved that the algorithm works for polygons known as weakly externally visible polygons.

In this paper a new notion called external left visibility is introduced and it is shown that this is a necessary and sufficient condition for the success of Sklansky's algorithm. Moreover, algorithms testing simple polygons for external left visibility and weak external visibility are given.     

Efficient planar convex hull algorithm

A new algorithm for computing the convex hull of a planar set of points is presented. The method of determining the set of suitable candidates is compared with previous algorithms. The algorithm is also compared with other algorithms in terms of running time and storage requirements, and experiments indicate that it is at least as good in terms of space and generally better in terms of running time.     

A simple linear hidden-line algorithm for star-shaped polygons

A very simple, linear-running-time algorithm is presented for solving the hidden-line problem for star-shaped polygons. The algorithm first decomposes the visibility regions into edge-visible polygons and then solves the hidden-line problem for these simpler polygons. In addition to simplicity the algorithm possesses the virtue of affording a very easy proof of correctness. Some applications where this problem arises are mentioned.     

A fast convex hull algorithm with maximum inscribed circle affine transformation

This paper presents a fast convex hull algorithm for a large point set. The algorithm imitates the procedure of human visual attention derived in a psychological experiment. The merit of human visual attention is to neglect most inner points directly. The proposed algorithm achieves a significant saving in time and space in comparison with the two best convex hull algorithms mentioned in a latest review proposed by Chadnov and Skvortsov in 2004. Furthermore, we propose to use an affine transformation to solve the narrow shape problem for computing the convex hull faster.     

A simple linear algorithm for intersecting convex polygons

LetP andQ be two convex polygons withm andn vertices, respectively, which are specified by their cartesian coordinates in order. A simpleO(m+n) algorithm is presented for computing the intersection ofP andQ. Unlike previous algorithms, the new algorithm consists of a two-step combination of two simple algorithms for finding convex hulls and triangulations of polygons.     

Two parallel algorithms for the convex hull problem in a two dimensional space

Two parallel algorithms for determining the convex hull of a set of data points in two dimensional space are presented. Both are suitable for MIMD parallel systems. The first is based on the strategy of divide-and-conquer, in which some simplest convex-hulls are generated first and then the final convex hull of all points is achieved by the processes of merging 2 sub-convex hulls. The second algorithm is by the process of picking up the points that are necessarily in the convex hull and discarding the points that are definitely not in the convex hull. Experimental results on a MIMD parallel system of 4 processors are analysed and presented.