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24
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 34 (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...
Compaction and Separation Algorithms for NonConvex Polygons and Their Applications
 European Journal of Operations Research
, 1995
"... Given a two dimensional, nonoverlapping layout of convex and nonconvex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied "forces." We apply compaction to improve the material utilization of an already tightly packed layout. Compaction can be mo ..."
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Cited by 32 (9 self)
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Given a two dimensional, nonoverlapping layout of convex and nonconvex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied "forces." We apply compaction to improve the material utilization of an already tightly packed layout. Compaction can be modeled as a motion of the polygons that reduces the value of some functional on their positions. Optimal compaction, planning a motion that reaches a layout that has the global minimum functional value among all reachable layouts, is shown to be NPcomplete under certain assumptions. We first present a compaction algorithm based on existing physical simulation approaches. This algorithm uses a new velocitybased optimization model. Our experimental results reveal the limitation of physical simulation: even though our new model improves the running time of our algorithm over previous simulation algorithms, the algorithm still can not compact typical layouts of one hundred or more polygons in ...
Translational Polygon Containment and Minimal Enclosure using Geometric Algorithms and Mathematical Programming
, 1995
"... We present an algorithm for the twodimensional translational containment problem: find translations for k polygons (with up to m vertices each) which place them inside a polygonal container (with n vertices) without overlapping. The polygons and container may be nonconvex. The containment algorit ..."
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Cited by 24 (13 self)
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We present an algorithm for the twodimensional translational containment problem: find translations for k polygons (with up to m vertices each) which place them inside a polygonal container (with n vertices) without overlapping. The polygons and container may be nonconvex. The containment algorithm consists of new algorithms for restriction, evaluation, and subdivision of twodimensional configuration spaces. The restriction and evaluation algorithms both depend heavily on linear programming; hence we call our algorithm an LP containment algorithm. Our LP containment algorithm is distinguished from previous containment algorithms by the way in which it applies principles of mathematical programming and also by its tight coupling of the evaluation and subdivision algorithms. Our new evaluation algorithm finds a local overlap minimum. Our distancebased subdivision algorithm eliminates a "false" (local but not global) overlap minimum and all layouts near that overlap minimum, a...
Multiple Translational Containment Part I: An Approximate Algorithm
 Algorithmica, special issue on Computational
, 1996
"... In Part I we present an algorithm for finding a solution to the twodimensional translational approximate multiple containment problem: find translations for k polygons which place them inside a polygonal container so that no point of any polygon is more than 2ffl inside of the boundary of any other ..."
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Cited by 17 (9 self)
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In Part I we present an algorithm for finding a solution to the twodimensional translational approximate multiple containment problem: find translations for k polygons which place them inside a polygonal container so that no point of any polygon is more than 2ffl inside of the boundary of any other polygon. The polygons and container may be nonconvex. The value of ffl is an input to the algorithm. In industrial applications, the containment solution acts as a guide to a machine cutting out polygonal shapes from a sheet of material. If one chooses ffl to be a fraction of the cutter's accuracy, then the solution to the approximate containment problem is sufficient for industrial purposes. Given a containment problem, we characterize its solution and create a collection of containment subproblems from this characterization. We solve each subproblem by first restricting certain twodimensional configuration spaces until a steady state is reached, and then testing for a solution inside the...
PositionBased Physics: Simulating the Motion of Many Highly Interacting Spheres and Polyhedra
 In Computer Graphics
, 1996
"... This paper proposes a simplified positionbased physics that allows us to rapidly generate "piles" or "clumps" of many objects: local energy minima under a variety of potential energy functions. We can also generate plausible motions for many highly interacting objects from arbitrary starting positi ..."
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Cited by 16 (4 self)
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This paper proposes a simplified positionbased physics that allows us to rapidly generate "piles" or "clumps" of many objects: local energy minima under a variety of potential energy functions. We can also generate plausible motions for many highly interacting objects from arbitrary starting positions to a local energy minimum. We present an efficient and numerically stable algorithm for carrying out positionbased physics on spheres and nonrotating polyhedra through the use of linear programming. This algorithm is a generalization of an algorithm for finding tight packings of (nonrotating) polygons in two dimensions. This work introduces linear programming as a useful tool for graphics animation. As its name implies, positionbased physics does not contain a notion of velocity, and thus it is not suitable for simulating the motion of freeflying, unencumbered objects. However, it generates realistic motions of "crowded" sets of objects in confined spaces, and it does so at least two...
Constraintbased Automatic Placement for Scene Composition
 IN GRAPHICS INTERFACE
, 2002
"... The layout of large scenes can be a timeconsuming and tedious task. In most current systems, the user must position each of the objects by hand, one at a time. This paper presents a constraintbased automatic placement system, which allows the user to quickly and easily lay out complex scenes. The ..."
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Cited by 16 (0 self)
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The layout of large scenes can be a timeconsuming and tedious task. In most current systems, the user must position each of the objects by hand, one at a time. This paper presents a constraintbased automatic placement system, which allows the user to quickly and easily lay out complex scenes. The system
Multiple Translational Containment Part II: Exact Algorithms
, 1994
"... We present exact algorithms for finding a solution to the twodimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. The term kCN denotes the version in which the polygons are convex and the container is non ..."
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Cited by 13 (7 self)
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We present exact algorithms for finding a solution to the twodimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. The term kCN denotes the version in which the polygons are convex and the container is nonconvex, and the term kNN denotes the version in which the polygons and the container are nonconvex. The notation (r; k)CN, (r; k)NN, and so forth refers to the problem of finding all subsets of size k out of r objects that can be placed in a container. The polygons have up to m vertices, and the container has n vertices, where n is usually much larger than m. We present exact algorithms for the following: 2CN in O(mn log n) time, (r; 2)CN in O(r 2 m log n) time (for r ?? n), 3CN in O(m 3 n log n) time, kCN in O(m 2k n k log n) or O((mn) k+1 ) time, and kNN in O((mn) 2k+1 LP(2k; 2k(2k + 1)mn + k(k \Gamma 1)m 2 )) time, where LP(a; b) is the time to solve a linear program with...
Multiple Containment Methods
, 1994
"... We present three different methods for finding solutions to the 2D translationonly containment problem: find translations for k polygons that place them inside a given polygonal container without overlap. Both the container and the polygons to be placed in it may be nonconvex. First, we provide se ..."
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Cited by 13 (11 self)
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We present three different methods for finding solutions to the 2D translationonly containment problem: find translations for k polygons that place them inside a given polygonal container without overlap. Both the container and the polygons to be placed in it may be nonconvex. First, we provide several exact algorithms that improve results for k = 2 or k = 3. In particular, we give an algorithm for three convex polygons and a nonconvex container with running time in O(m 3 n log mn), where n is the number of vertices in the container, and m is the sum of the vertices of the k polygons. This is an improvement of a factor of n 2 over previous algorithms. Second, we give an approximation algorithm for k nonconvex polygons and a nonconvex container based on restriction and subdivision of the configuration space. Third, we develop a MIP (mixed integer programming) model for k nonconvex polygons and a nonconvex container.
Containment Algorithms for Nonconvex Polygons with Applications to Layout
, 1995
"... Layout and packing are NPhard geometric optimization problems of practical importance for which finding a globally optimal solution is intractable if P!=NP. Such problems appear in industries such as aerospace, ship building, apparel and shoe manufacturing, furniture production, and steel construct ..."
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Cited by 13 (6 self)
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Layout and packing are NPhard geometric optimization problems of practical importance for which finding a globally optimal solution is intractable if P!=NP. Such problems appear in industries such as aerospace, ship building, apparel and shoe manufacturing, furniture production, and steel construction. At their core, layout and packing problems have the common geometric feasibility problem of containment: find a way of placing a set of items into a container. In this thesis, we focus on containment and its applications to layout and packing problems. We demonstrate that, although containment is NPhard, it is fruitful to: 1) develop algorithms for containment, as opposed to heuristics, 2) design containment algorithms so that they say "no" almost as fast as they say "yes", 3) use geometric techniques, not just mathematical programming techniques, and 4) maximize the number of items for which the algorithms are practical. Our approach to containment is based on a new restrict/evaluate...
Multiple Translational Containment: Approximate and Exact Algorithms
, 1995
"... We present exact algorithms for finding a solution to the twodimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. We also give an approximate algorithm: given any ffl, it finds a set of translations such t ..."
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Cited by 13 (7 self)
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We present exact algorithms for finding a solution to the twodimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. We also give an approximate algorithm: given any ffl, it finds a set of translations such that no point of any polygon is more than 2ffl inside the boundary of any other polygon or outside the container. The term kCN denotes a containment problem in which the k polygons are convex and the container is nonconvex, and kNN denotes nonconvex polygons and container. The polygons have up to m vertices, and the container has n vertices, where n ? m (typically). We give exact algorithms for the following: 2CN in O(mn log n) time, 3CN in O(m 3 n log n) time, and kNN in O((mn) 2k+1 LP(2k; 2kmn + k 2 m 2 )) time, where LP(a; b) is the time to solve a linear program with a variables and b constraints. We present an approximate algorithm for kNN whose running time is O( \Gamma 1 ...