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147
Maximizing the Overlap of Two Planar Convex Sets under Rigid Motions
, 2006
"... Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε> 0, we compute a rigid motion such that the area of overlap is at least 1−ε times the maximum possible overlap. Our algorith ..."
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Cited by 12 (5 self)
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Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε> 0, we compute a rigid motion such that the area of overlap is at least 1−ε times the maximum possible overlap. Our algorithm uses O(1/ε) extreme point and line intersection queries on P and Q, plus O((1/ε 2) log(1/ε)) running time. If only translations are allowed, the extra running time reduces to O((1/ε) log(1/ε)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/ε) log n + (1/ε 2) log(1/ε)) for rigid motions and O((1/ε) log n + (1/ε) log(1/ε)) for translations.
Reactive Robotics I: Reactive Grasping with a Modified Gripper and Multifingered Hands
 The International Journal of Robotics Research
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Characterizing and Efficiently Computing Quadrangulations of Planar Point Sets
, 1997
"... Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (exceptpossibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and o ..."
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Cited by 10 (2 self)
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Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (exceptpossibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If Sdoes not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an \Omega (n log n) time lower bound for the problem. Our results imply that a kangulation of a set of points can be achieved with the addition of at most k \Gamma 3extra points within the same time bound. Finally, we present an experimental comparison between three quadrangulation algorithms which shows that the Spiraling Rotating Calipers (SRC) algorithm(presented in Section 4) produces quadrangulations with the greatest number of convex quadrilaterals as well as those with the smallest difference between the average minimum and maximum angle over all quadrangles.
Translating a Convex Polygon to Contain a Maximum Number of Points (Extended Abstract)
 PROC. 7TH CANADIAN CONF. ON COMPUTATIONAL GEOMETRY
, 1995
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A Convexity Measurement for Polygons
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2002
"... Convexity estimators are commonly used in the analysis of shape. In this paper we define and evaluate a new easily computable measure of convexity for polygons. Let P be an arbitrary polygon. If 7 ) (P, c) denotes the perimeter in the sense of l metrics of the polygon obtained by the rotation of ..."
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Cited by 10 (3 self)
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Convexity estimators are commonly used in the analysis of shape. In this paper we define and evaluate a new easily computable measure of convexity for polygons. Let P be an arbitrary polygon. If 7 ) (P, c) denotes the perimeter in the sense of l metrics of the polygon obtained by the rotation of P by angle c with the origin as the center of the applied rotation, and if 7)2 (R(P, c)) is the Euclidean perimeter of the minimal rectangle R(P, c) having the edges parallel to coordinate axes which includes such a rotated polygon P, then we show that C(P) defined as C(P) = min 7)2(R(P,c)) a C[0,2rr] 7)1 (P, c) can be used as an estimate for the convexity of P. Several desirable properties of C (P) are proved, as well.
Polygon Containment And Translational MinHausdorffDistance Between Segment Sets Are
"... The 3sum problem represents a class of problems conjectured to require\Omega\Gamma n 2 ) time to solve, where n is the size of the input. Given two polygons P and Q in the plane, we show that some variants of the decision problem, whether there exists a transformation of P that makes it conta ..."
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Cited by 10 (1 self)
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The 3sum problem represents a class of problems conjectured to require\Omega\Gamma n 2 ) time to solve, where n is the size of the input. Given two polygons P and Q in the plane, we show that some variants of the decision problem, whether there exists a transformation of P that makes it contained in Q, are 3sumhard. In the first variant P and Q are any simple polygons and the allowed transformations are translations only; in the second and third variants both polygons are convex and we allow either rotations only or any rigid motion. We also show that finding the translation in the plane that minimizes the Hausdorff distance between two segment sets is 3sumhard. Keywords: Computational complexity, 3sumhardness, polygon containment, Hausdorff distance, segment sets. 1.
Optimal Placement of Convex Polygons to Maximize Point Containment
, 1996
"... Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the co ..."
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Cited by 8 (1 self)
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Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the containing regions in the parameter space of transformations. The algorithm requires O(nk 2 m 2 log(mk)) time and O(n +m) space, where k is the maximum number of points contained. This provides a linear improvement over the best previously known algorithm when k is large (\Theta(n)) and a cubic improvement when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. 1 Introduction A planar rigid motion ae is an affine transformation of the plane that preserves distance (and therefore angles and area also). We say that a polygon P contains a set S of points if every point in S lies on P or in the interior of P . In th...
Constructing optimal highways
, 2010
"... For two points p and q in the plane, a straight line h, called a highway, and a real v> 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed v on h and with speed 1 in t ..."
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Cited by 8 (2 self)
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For two points p and q in the plane, a straight line h, called a highway, and a real v> 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed v on h and with speed 1 in the underlying metric elsewhere. Given a set S of n points in the plane and a highway speed v, we consider the problem of finding a highway that minimizes the maximum travel time over all pairs of points in S. If the orientation of the highway is fixed, the optimal highway can be computed in linear time, both for the L1 and the Euclidean metric as the underlying metric. If arbitrary orientations are allowed, then the optimal highway can be computed in O(n 2 log n) time. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.
On Separating Two Simple Polygons by a Single Translation
 Discrete Computational Geometry
, 1989
"... Let P and Q be two disjoint simple polygons having n sides each. We present an algorithm which determines whether Q can be moved by a single translation to a position sufficiently far from P, and which produces all such motions if they exist. The algorithm runs in time O(t(n)) where t(n) is the time ..."
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Cited by 8 (0 self)
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Let P and Q be two disjoint simple polygons having n sides each. We present an algorithm which determines whether Q can be moved by a single translation to a position sufficiently far from P, and which produces all such motions if they exist. The algorithm runs in time O(t(n)) where t(n) is the time needed to triangulate an nsided polygon. Since Tarjan and Van Wyk have recently shown that t(n) = O(n log log n) this improves the previous best result for this problem which was O(n log n) even after triangulation. 1. Introduction Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout considerable attention has been devoted recently to the problem of moving polygons in the plane without collisions [1][11]. A typical problem in robotics is the FINDPATH problem [12], where a robot must determine if an object, modeled as a polygon in the plane, can be moved from a starting position to a goal state without collisions occurring between the object being m...
Approximating Parametric Curves with Strip Trees using Affine Arithmetic
"... We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance ..."
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Cited by 8 (3 self)
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We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance fields for parametric curves.