We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with no self-intersections are NP-hard.

Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, including rectangles and halfspaces. In addition, we give extensions to other discrepancy problems. 1. Introduction The main theme of this paper is to present efficient algorithms that solve the problem of computing the maximum bichromatic discrepancy for axis oriented rectangles. This problem arises naturally in different areas of computer science, such as computational 1 The research work of these authors was supported by NSF Grant CCR93-01254 and the Geometry Center. learning theory, computational geometry and computer graphics ([Ma], [DG]), and has applications in all these areas. In computational learning theory, the problem of agnostic PAC-learning with simple geometric hypothese...

A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

Determining the convex hull of a point set is a basic operation for many applications of pattern recognition, image processing, statistics, and data mining.

The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes linear space and allows for reporting the approximation with relative accuracy ffl in O( p 1=ffl log n) time; and the update time is O(log 2 n). The method uses the tentative prune-and-search strategy of Kirkpatrick and Snoeyink.

An in-place algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. In this paper we describe three in-place algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others...

We show that many classical problems of Computational Geometry can be solved in the cylinder, but planar techniques cannot be adapted always successfully and new techniques must be considered. 1 Introduction and preliminaries Some computational problems that are inherently geometrical in nature have their input and output data in some spaces that are neither the plane nor the Euclidean space (we will call to the plane and the Euclidean space, the Euclidean cases), but Computational Geometry is mostly concerned with Euclidean cases (of course, there are some exceptions as we will see later). In fact, frequently it is assumed that planar algorithms are valid, in general, if they are translated, for instance, to the cylinder, but, in practice, many problems arise in that translation. It is the intention of this work to demonstrate that classical problems of Computational Geometry can be solved when the input and output data are on the cylinder, but that planar techniques cannot be adapt...

We present the first part of a study on what we call quality pictures, where we introduce a quality parameter ff that indicates the minimum incidence angle allowed between the vision direction and a seen surface. We solve here some combinatorial and algorithmic problems about both external and internal quality pictures of polygons. 1 Introduction Since 1973, when V. Klee proposed the problem of determining the minimum number of points (guards or cameras) that would always suffice to see any n-polygon, many variations of this problem have been studied [20], [22]. Various studies have considered restricting the class of polygons to be guarded (orthogonal, etc.), new kinds of guards have been introduced (edgeguards, mobile guards,...) and even watchman routes have been studied. Traditionally, it is assumed that guards can see in any direction. If we imagine cameras, instead of guards, this would imply a 360 ffi field of aperture. More recently, more realistic approaches have been propo...

Advanced computer graphics techniques such as ray tracing... In this paper, we provide algorithms and geometries providing optimal mirror placements for two natural classes of problems; object-independent and object-dependent mirror placement. In the object-independent formulation, we seek to design a "camera" with a fixed geometry of mirrors and viewpoint which satisfies certain criteria for all objects in a specific class. The object-independent formulation is well-suited to visualizing scientific data sets, since extensive analysis prior to an initial rendering may be expensive or (as in the case of volumetric data without an explicit geometric model) impossible. In Section 2, we present the design of a camera which guarantees complete visibility for any orientation of any convex object in both 2 and 3 dimensions using the minimum number of mirrors, and which also minimizes the maximum distance between a point in the object and its (virtual) viewpoint. This distance criteria is important because it ensures that the reflected images are as high-resolution as possible...

Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, including rectangles and halfspaces. In addition, we give extensions to other discrepancy problems. 1 Introduction The main theme of this paper is to present efficient algorithms that solve the problem of computing the maximum bichromatic discrepancy for axis oriented rectangles. This problem arises naturally in different areas of computer science, such as computational learning theory, computational geometry and computer graphics ([Ma], [DG]), and has applications in all these areas. In computational learning theory, the problem of agnostic PAC-learning with simple geometric hypotheses can be reduced to the problem of computing the maximum bichromatic discrepancy for simple geometric ra...