Victor Milenkovic

Compaction and separation algorithms for non-convex polygons and their applications

Abstract Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons resulting from applied ‘forces’. By moving many polygons simultaneously, compaction can improve the material utilization of even tightly packed layouts. Compaction is hard: finding the tightest layout that a valid motion can reach is PSPACE-hard, and even compacting to a local optimum can require an exponential number of moves. Our first compaction algorithm uses a new velocity-based optimization model based on existing physical simulation approaches. The performance of this algorithm illustrates that these approaches cannot quickly compact tightly packed layouts. We then present a new position-based optimization model. This model represents the forces as a linear objective function, and it permits direct calculation, via linear programming, of new non-overlapping polygon positions at a local minimum of the objective. The new model yields a translational compaction algorithm that runs two orders of magnitude faster than physical simulation methods. We also consider the problem of separating overlapping polygons using the least amount of motion, prove optimal separation to be NP-complete, and show that our position-based model also yields an efficient algorithm for finding a locally optimal separation. The compaction algorithm has improved cloth utilization of human-generated layouts of trousers and other garments. Given a database of human-generated markers and with an efficient technique for matching, the separation algorithm can automatically generate layouts that approach human performance.

Verifiable implementation of geometric algorithms using finite precision arithmetic

Abstract Two methods are proposed for correct and verifiable geometric reasoning using finite precision arithmetic. The first method, data normalization, transforms the geometric structure into a configuration for which all finite precision calculations yield correct answers. The second method, called the hidden variable method, constructs configurations that belong to objects in an infinite precision domain—without actually representing these infinite precision objects. Data normalization is applied to the problem of modeling polygonal regions in the plane, and the hidden variable method is used to calculate arrangements of lines.

Optimization-based animation

Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex three-dimensional bodies settling into stacks and other “crowded” arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta and accelerations strictly at frame times. Contact points are synchronized at the end of each frame. The extremely small integration steps inherent to traditional simulation techniques are avoided. Non-convex bodies are simulated as unions of convex bodies. Links and joints are simulated successfully with bi-directional constraints. A hybrid of OBA and retroactive detection (RD) has been implemented as well. A review of existing work finds no other packages that can simulate similarly complex scenes in a practical amount of time.

Double precision geometry: a general technique for calculating line and segment intersections using rounded arithmetic

For the first time it is shown how to reduce the cost of performing specific geometric constructions by using rounded arithmetic instead of exact arithmetic. By exploiting a property of floating-point arithmetic called monotonicity, a technique called double-precision geometry can replace exact arithmetic with rounded arithmetic in any efficient algorithm for computing the set of intersections of a set of lines or line segments. The technique reduces the complexity of any such line or segment arrangement algorithm by a constant factor. In addition, double-precision geometry reduces by a factor of N the complexity of rendering segment arrangements on a 2/sup N/*2/sup N/ integer grid such that output segments have grid points as endpoints.<<ETX>>

Finding the largest area axis-parallel rectangle in a polygon

Abstract This paper considers the geometric optimization problem of finding the Largest area axis-parallel Rectangle (LR) in an n-vertex general polygon. We characterize the LR for general polygons by considering different cases based on the types of contacts between the rectangle and the polygon. A general framework is presented for solving a key subproblem of the LR problem which dominates the running time for a variety of polygon types. This framework permits us to transform an algorithm for orthogonal polygons into an algorithm for non-orthogonal polygons. Using this framework, we show that the LR in a general polygon (allowing holes) can be found in O(n log2 n) time. This matches the running time of the best known algorithm for orthogonal polygons. References are given for the application of the framework to other types of polygons. For each type, the running time of the resulting algorithm matches the running time of the best known algorithm for orthogonal polygons of that type. A lower bound of time in Ω(n log n) is established for finding the LR in both self-intersecting polygons and general polygons with holes. The latter result gives us both a lower bound of Ω(n log n) and an upper bound of O(n log2 n) for general polygons.

Rotational polygon containment and minimum enclosure using only robust 2D constructions

Abstract An algorithm and a robust floating point implementation is given for rotational polygon containment: given polygons P1,P2,P3,…,Pk 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: given a class C of container polygons, find a container C∈ C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1−e)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 intersection: the intersection of a polygon with itself as it rotates through a range of angles. In particular, it introduces nearest pair rounding, which allows all these calculations to be carried out in rounded floating point arithmetic.

Numerical stability of algorithms for line arrangements

We analyze the behavior of two line arrangement algorithms, a sweepline algorithm and an incremental algorithm, in approximate arithmetic. The algorithms have running times O(n2 log n) and O(n2 ) respectively. We show that each of these algorithms can be implemented to have O(nc) relative error. This means that each algorithm produces an arrangement realized by a set of pseudolines so that each pseudoline differs from the corresponding line relatively by at most 0( nc). We also show that there is a line arrangement algorithm with 0(n2 log n) running time and O(c) relative error.

Shortest Path Geometric Rounding

Abstract. 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 significant 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 combinatorial change and geometric error than the other rounding methods. Three algorithms are given for shortest path rounding, one of which we have used in industrial application software since 1992. In combination with recent advances in exact floating point evaluation of numerical primitives, shortest path geometric rounding yields a practical solution to numerical issues in computational geometry. Geometric algorithms can be implemented exactly on floating point input coordinates; the exact output coordinates can be rounded to accurate floating point approximations; and the cost of each arithmetic operation is only a little more than if it were implemented as a single hardware floating point operation.

Rotational polygon overlap minimization and compaction

Abstract 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 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.

Multiple Translational Containment Part I: An Approximate Algorithm

Abstract. 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 2ε inside of the boundary of any other polygon. The polygons and container may be nonconvex. The value of ε 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 ε is chosen to be a fraction of the cutters 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 configuration spaces. If necessary, we subdivide the configuration spaces to generate new subproblems. The running time of our algorithm is