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 (1-epsilon)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...

We present an algorithm for the two-dimensional 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 two-dimensional 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 distance-based subdivision algorithm eliminates a "false" (local but not global) overlap minimum and all layouts near that overlap minimum, a...

In Part I we present an algorithm for finding a solution to the two-dimensional 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 two-dimensional configuration spaces until a steady state is reached, and then testing for a solution inside the...

Exact implementations of algorithms of computational geometry are subject to exponential growth in running time and space. In particular, coordinate bit-complexity can grow exponentially when algorithms are cascaded: the output of one algorithm becomes the input to the next. Cascading is a signicant problem in practice. We propose a geometric rounding technique: shortest path rounding. Shortest path rounding trades accuracy for space and time and eliminates the exponential cost introduced by cascading. It can be applied to all algorithms which operate on planar polygonal regions, for example, set operations, transformations, convex hull, triangulation, and Minkowski sum. Unlike other geometric rounding techniques, shortest path rounding can round vertices to arbitrary lattices, even in polar coordinates, as long as the rounding cells are connected. (Other rounding techniques can only round to the integer grid.) On the integer grid, shortest path rounding introduces less com...

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 non-overlapping layout in a rectangular container, plan a non-overlapping 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...

A translation lattice packing of k polygons P 1 ; P 2 ; P 3 ; : : : ; P k is a (non-overlapping) packing of the k polygons which can be replicated without overlap at each point of a lattice i 0 v 0 + i 1 v 1 , where v 0 and v 1 are vectors generating the lattice and i 0 and i 1 range over all integers. A densest translational lattice packing is one which minimizes the area jv 0 v 1 j of the fundamental parallelogram. An algorithm and implementation is given for densest translation lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture. 1 Introduction A number of industries generate new parts by cutting them from stock material: cloth, leather (hides), sheet metal, glass, etc. These industries need to generate dense non-overlapping layouts of polygonal shapes. Because fabric has a grain, apparel layouts usually permit only a nite set of orientations. Since cloth comes in rolls, the most common layout problem in the apparel industr...

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 non-overlapping layout in a rectangular container, plan a non-overlapping motion that minimizes the length or area of the container. Containment: place the polygons into a (possibly non-convex container) without overlapping. Minimal enclosure: find a non-overlapping layout inside a minimum-length, fixed-width rectangle or inside a minimum area rectangle. All of these algorithms have important i...

The oriented strip packing problem is very important to manufacturing industries: given a strip of fixed width and a set of many (? 100) nonconvex polygons with 1, 2, 4, or 8 orientations permitted for each polygon, find a set of translations and orientations for the polygons that places them without overlapping into the strip of minimum length. Two heuristics are given for strip packing: 1) for the translation-only version and 2) for the oriented version. The first heuristic uses an algorithm we have previously developed for translational containment: given polygons P 1 ; P 2 ; : : : ; P k and a fixed container C, find translations for the polygons that place them into C without overlapping. The containment algorithm is practical for k 10. Two new containment algorithms are presented for use in the second packing heuristic. The first, an ordered containment algorithm, solves containment in time which is only linear in k when the polygons are a) "long" with respect to one dimension o...

This paper deals with cutting/packing problems in which there is a set of pieces to be allocated and arranged in a set of "containers." In an apparel manufacturing application, the containers might be unused areas of the fabric after large pieces have been placed, and the pieces of interest might be the smaller pieces. In a sheet metal application, the containers could be the sheets themselves, and the pieces the entire set of pieces to be arranged. The specific problem addressed takes as input a set of groups (of pieces), and mappings from pieces to groups and groups to containers. The method in which the groups are generated and the particular geometric constraints (e.g., translation only, or translation plus rotation) is not critical for the methods developed here. This paper presents an integer programming formulation of the multiple-container group assignment problem (MCGAP). Based on long and/or highly variable solution times for some problem instances, a Lagrangian heuristic pro...

The footwear industry's need for an automatic containment algorithm is becoming increasingly important within the manufacturing process. Irregular containers, such as hides, have many different quality regions and holes that must be taken into account when containment is done because they represent an important cost. Automatic containment processes should be aware of these factors and still perform in practical time. We present an iterative containment algorithm that uses Minkowski operators and can be applicable to such containment problems. Although the iterative solution is not the optimal one, it can reach a solution in practical running times and it can get results that approximate the human made containment process. Keywords: Containment Problems, Minkowski Operators, Evaluators.