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Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 114 (10 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 81 (17 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Rotational polygon containment and minimum enclosure using only robust 2D constructions
 Computational Geometry
, 1998
"... An algorithm and a robust floating point implementation is given for rotational polygon containment:given polygons P 1 ,P 2 ,P 3 ,...,P k and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm a ..."
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Cited by 36 (6 self)
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An algorithm and a robust floating point implementation is given for rotational polygon containment:given polygons P 1 ,P 2 ,P 3 ,...,P k and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: givenaclass C of container polygons, find a container C in C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1epsilon)A and A. Experiments indicate that finding the minimum enclosure is practical for k = 2, 3 but not larger unless optimality is sacrificed or angles ranges are limited (although these solutions can still be useful). Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range in...
Motion Planning for a Convex Polygon in a Polygonal Environment
 Geom
, 1997
"... We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which i ..."
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Cited by 17 (8 self)
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We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which is nearly optimal in the worst case. The algorithm is also conceptually relatively simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also nearquadratic in mn. In addition, we describe an algorithm that preprocesses the computed free configuration space so that `reachability' queries can be answered in polylogarithmic time. All three authors have been supported by a grant from the U.S.Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Gr...
Generalized penetration depth computation
 In SPM ’06: Proceedings of the 2006 ACM symposium on Solid and physical modeling
, 2006
"... Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must ..."
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Cited by 11 (2 self)
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Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are nonconvex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path nonexistence.
Rotational Polygon Overlap Minimization and Compaction
 Computational Geometry: Theory and Applications
, 1998
"... An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P 1 ,P 2 ,P 3 ,...,P k in a container polygon Q, translate and rotate the polygons to diminish their overlap to a local minimum. A (local) overlap minimum has the property that any p ..."
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Cited by 9 (2 self)
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An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P 1 ,P 2 ,P 3 ,...,P k in a container polygon Q, translate and rotate the polygons to diminish their overlap to a local minimum. A (local) overlap minimum has the property that any perturbation of the polygons increases the overlap. Overlap minimization is modified to create a practical algorithm for compaction: starting with a nonoverlapping layout in a rectangular container, plan a nonoverlapping motion that diminishes the length or area of the container to a local minimum. Experiments show that both overlap minimization and compaction work well in practice and are likely to be useful in industrial applications. 1998 Published by Elsevier Science B.V. Keywords: Layout; Packing or nesting of irregular polygons; Containment; Minimum enclosure; Compaction; Linear programming 1. Introduction A number of industries generate new parts by cutting them from stock mater...
Rotational Polygon Overlap Minimization
 Computational Geometry: Theory and Applications
, 1997
"... An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P1 ; P2 ; P3 ; : : : ; Pk in a container polygon C, translate and rotate the polygons to a layout that minimizes an overlap measure. A (local) overlap minimum has the property that ..."
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Cited by 5 (1 self)
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An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P1 ; P2 ; P3 ; : : : ; Pk in a container polygon C, translate and rotate the polygons to a layout that minimizes an overlap measure. A (local) overlap minimum has the property that any perturbation of the polygons increases the chosen measure of overlap. Experiments show that the algorithm works well in practice. It is shown how to apply overlap minimization to create algorithms for other layout tasks: compaction, containment, and minimal enclosure. Compaction: starting with a nonoverlapping layout in a rectangular container, plan a nonoverlapping motion that minimizes the length or area of the container. Containment: place the polygons into a (possibly nonconvex container) without overlapping. Minimal enclosure: find a nonoverlapping layout inside a minimumlength, fixedwidth rectangle or inside a minimum area rectangle. All of these algorithms have important i...
A Geometric Framework for Solving Subsequence Problems in Computational Biology Efficiently
"... In this paper, we introduce the notion of a constrained Minkowski sum which for two (finite) pointsets P,Q ⊆ R 2 and a set of k inequalities Ax � b is defined as the pointset (P ⊕ Q)Ax�b = {x = p+ q  p ∈ P, q ∈ Q, Ax � b}. We show that typical subsequence problems from computational biology can be ..."
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Cited by 5 (0 self)
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In this paper, we introduce the notion of a constrained Minkowski sum which for two (finite) pointsets P,Q ⊆ R 2 and a set of k inequalities Ax � b is defined as the pointset (P ⊕ Q)Ax�b = {x = p+ q  p ∈ P, q ∈ Q, Ax � b}. We show that typical subsequence problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O(N log N), where N = P+Q  if k is fixed. For the special case (P ⊕ Q)x1�β, where P and Q consist of points with integer x1coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many subsequence problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of subsequence problems can be modeled and solved. 1
Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently
"... In this paper, we introduce the notion of a constrained Minkowski sum which for two (finite) pointsets P,Q ⊆ R 2 and a set of k inequalities Ax � b is defined as the pointset (P ⊕ Q)Ax�b = {x = p+q  p ∈ P, q ∈ Q, Ax � b}. We show that typical interval problems from computational biology can be so ..."
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In this paper, we introduce the notion of a constrained Minkowski sum which for two (finite) pointsets P,Q ⊆ R 2 and a set of k inequalities Ax � b is defined as the pointset (P ⊕ Q)Ax�b = {x = p+q  p ∈ P, q ∈ Q, Ax � b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O(N logN), where N = P+Q  if k is fixed. For the special case (P ⊕ Q) x1�β, where P and Q consist of points with integer x1coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many interval problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of interval problems can be modeled and solved. 1
Some Regularity Measures for Convex Polygons
"... We propose new measures to evaluate to which extent the shape of a given convex polygon is close to the shape of some regular polygon. We prove that our parameters satisfy several reasonable requirements and provide algorithms for their efficient computation. The properties we mostly focus on are th ..."
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Cited by 1 (1 self)
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We propose new measures to evaluate to which extent the shape of a given convex polygon is close to the shape of some regular polygon. We prove that our parameters satisfy several reasonable requirements and provide algorithms for their efficient computation. The properties we mostly focus on are the facts that regular polygons are equilateral, equiangular and have radial symmetry. 1