Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimum-area rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vector-sum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 1.

Abstract-Let S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estimators in statistics. It also provides valuable information on the mor-phology of a set of sites and has proven to be an efficient preconditioning for range search problems. An optimal algorithm is described for comput-ing the convex layers of S. The algorithm runs in O ( n log n) time and requires O(n) space. Also addressed is the problem of determining the depth of a query point within the convex layers of S, i.e., the number of layers that enclose the query point. This is essentially a planar point location problem, for which optimal solutions are therefore known. Taking advantage of structural properties of the problem, however, a much simpler optimal solution is derived. L I.

This paper describes a new method for triangulating a simple n-sided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of s is in practice a very small constant, even for extremely winding polygons. Our algorithm is the first method whose performance is linear in the number of vertices, up to within a factor that depends only on the shape-complexity of the polygon. Informally, this notion of shape-complexity measures how entangled a polygon is, and is thus highly independent of the number of vertices. A practical advantage of the algorithm is that it does not require sorting or the use of any balanced tree structure. Aside from the notion of sinuosity, we are also able to characterize a large class of polygons for which the algorithm can be proven to run in O(n log log n) time. The algorithm has been implemented, tested, and empirical evidence has confirmed its theoretical claim to efficiency.

In many computer applications areas such as graphics, automated cartography, image processing, and robotics the notion of visibility among objects modeled as polygons is a recurring theme. This paper is concerned with the visibility of a simple polygon from one of its edges. Three natural definitions of the visibility of a polygon from an edge are presented. The following computational problem is considered. Given an n- sided simple polygon, is the polygon visible from a specified edge? An O(n), and thus optimal, algorithm is exhibited for determining edge visibility under any of the three definitions. The paper closes with an interesting characterization of visibility and some open problems in this area. Index Terms - Algorithms, computational complexity, computational geometry, computer graphics, hidden line problems, image processing, robotics, simple polygon, visibility. 1. Introduction The notion of visibility in geometric objects is one that appears in many applications: the ...

Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, present new results, and provides a list of open problems and suggestions for further research. Key Words and Phrases: sofa problem, polygons, polyhedra, movable separability, visibility hulls, hidden lines, hidden surfaces, algorithms, complexity, computational geometry, spatial planning, collision avoidance, robotics, artificial intelligence. CR Categories: 3.36, 3.63, 5.25. 5.32. 5.5 * Research supported by NSERC Grant no. A9293 and FCAR Grant no.EQ1678. - 2 - ...

Given a graph G and a drawing or layout of G, it is sometimes desirable to alter or adjust the layout. The challenging aspect of designing layout adjustment algorithms is to maintain a user's mental picture of the original layout. We present a new approach to layout adjustment called cluster busting in anchored graph drawing. We then give two algorithms as examples of this approach. The goals of cluster busting in anchored graph drawing are to more evenly distribute the nodes of the graph in a drawing window while maintaining the user's mental picture of the original drawing. We present simple and eÆcient iterative heuristics to accomplish these goals. We formally de ne some measures of distribution and similarity and give empirical results based on these measures to quantify our methods. The theoretical analysis of our heuristics presents a formidable challenge, thus justifying our empirical analysis. Communicated by G. Di Battista: submitted April 1996; revised March 1998. Research...

Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon.) However, there are collections of line segments where this cannot be done. In this note it is proved that deciding whether a set of line segments admits a simple circuit is NP-complete. Deciding whether a set of horizontal line segments can be connected with horizontal and vertical line segments to construct an orthogonal simple circuit is also shown to be NP-complete.

We consider n independent points with a common but arbitrary density fin R a. Two points (X~, Xj) are joined by an edge when a certain set S(X~, Xj) does not contain any other data points. The expected number E(N) of edges in the graph depends upon n, f and the definition of S. Examples include rectangles, spheres and loons; these lead to the graph of all dominance pairs, the Gabriel graph and the relative neighborhood graph, respectively. Other graphs covered by our analysis include the nearest neighbor graph and the directional nearest neighbor graph. In all cases, we obtain asymptotic lower bounds that do not depend uponf(and are hence useful in all applications involving these graphs, since we usually do not know f). For sparse graphs, exact asymptotic constants are obtained for E(N) that are valid for all densities.

Digital images have become an important source of information in the modern world of communication systems. In their raw form, digital images require a tremendous amount of memory. Many research efforts have been devoted to the problem of image compression in the last two decades. Two different compression categories must be distinguished: lossless and lossy. Lossless compression is achieved if no distortion is introduced in the coded image. Applications requiring this type of compression include medical imaging and satellite photography. For applications such as video--telephony or multimedia applications some loss of information is usually tolerated in exchange for a high compression ratio.

In this paper we show how a theorem in plane geometry can be converted into an O(n log n) algorithm for decomposing a polygon into star-shaped subsets. The computational efficiency or this new decomposition contrasts with the heavy computational burden of existing methods. 1.0 Introduction The decomposition of a simple planar polygon into simpler components plays an important role in syntactic pattern recognition. Some examples of possible decompositions are decompositions into convex polygons [1], [2], decompositions into spiral polygons [3] and decompositions into monotone polygons [4]. A survey of these methods and many other additional references are contained in Pavlidis [5]. A star-shaped polygon is one in which the entire polygon is visible from at least one fixed point of the polygon. In this note we consider decompositions into star-shaped polygons and give an efficient algorithm for this problem. A similar decomposition has previously been suggested by Maruyama [6] in his th...