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12
Approximating Polygons and Subdivisions with MinimumLink Paths
, 1991
"... 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 object ..."
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Cited by 61 (11 self)
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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 selfintersections are NPhard.
Computing the Maximum Bichromatic Discrepancy, with applications to Computer Graphics and Machine Learning
 in Computer Graphics and Machine Learning. Journal of Computer and Systems Sciences
, 1996
"... 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, ..."
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Cited by 39 (8 self)
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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 CCR9301254 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 PAClearning with simple geometric hypothese...
Spaceefficient planar convex hull algorithms
 Proc. Latin American Theoretical Informatics
, 2002
"... A spaceefficient 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 spaceefficient algorithms for computing the convex hull of a planar point set. ..."
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Cited by 20 (1 self)
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A spaceefficient 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 spaceefficient algorithms for computing the convex hull of a planar point set.
Maintaining the Approximate Width of a Set of Points in the Plane (Extended Abstract)
, 1993
"... 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 lin ..."
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Cited by 12 (1 self)
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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(log² n). The method uses the tentative pruneandsearch strategy of Kirkpatrick and Snoeyink.
Determining the Convex Hull in Large Multidimensional Databases
, 2001
"... Determining the convex hull of a point set is a basic operation for many applications of pattern recognition, image processing, statistics, and data mining. ..."
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Cited by 12 (1 self)
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Determining the convex hull of a point set is a basic operation for many applications of pattern recognition, image processing, statistics, and data mining.
Optimal inplace planar convex hull algorithms
 Proceedings of Latin American Theoretical Informatics (LATIN 2002), volume 2286 of Lecture Notes in Computer Science
, 2002
"... An inplace 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 inplace algorithms for computing the convex hull of a planar point set. All three algorithms are optima ..."
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Cited by 5 (2 self)
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An inplace 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 inplace algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others...
Computation on parametric curves with an application in grasping
 International Journal of Robotics Research
"... Curved shapes are frequent subjects of maneuvers by the human hand. In robotics it is well known that antipodal grasps exist on curved objects and guarantee force closure under proper finger contact conditions. This paper presents an efficient algorithm that computes, up to numerical resolution, all ..."
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Cited by 2 (1 self)
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Curved shapes are frequent subjects of maneuvers by the human hand. In robotics it is well known that antipodal grasps exist on curved objects and guarantee force closure under proper finger contact conditions. This paper presents an efficient algorithm that computes, up to numerical resolution, all pairs of antipodal points on a simple, closed, and twice continuously differentiable plane curve. Dissecting the curve into segments everywhere convex or everywhere concave, the algorithm marches simultaneously on a pair of such segments with provable convergence and interleaves marching with numerical bisection recursively. It makes use of new insights into the differential geometry at two antipodal points. We have avoided resorting to traditional nonlinear programming which would neither be quite as efficient nor guarantee to find all antipodal points. A byproduct of our result is a procedure that constructs all common tangent lines of two curves, achieving quadratic convergence rate. Dissection and the coupling of marching with bisection constitute an algorithm design scheme potentially applicable to computational problems involving curves and curved shapes. KEY WORDS—antipodal point, antipodal angle, inflection, monotonicity, common tangent, convergence rate, robot grasping 1
Computational Geometry on the Cylinder (Extended Abstract)
"... 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 hav ..."
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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...
Incidence Angle Constrained Visibility
, 1996
"... 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 inter ..."
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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 npolygon, 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...
Illuminating Objects with Mirrors
, 1998
"... 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; objectindependent and objectdependent mirror placement. In the objectindependent formulation, we seek to design ..."
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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; objectindependent and objectdependent mirror placement. In the objectindependent 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 objectindependent formulation is wellsuited 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 highresolution as possible...