a b c d e f Figure 1: By overwriting voronoi regions, tile centroids are displaced away from an edge. Recentering tiles at their new centroids eventually moves them clear of the edge. This paper presents a method for simulating decorative tile mosaics. Such mosaics are challenging because the square tiles that comprise them must be packed tightly and yet must follow orientations chosen by the artist. Based on an existing image and user-selected edge features, the method can both reproduce the image’s colours and emphasize the selected edges by placing tiles that follow the edges. The method uses centroidal voronoi diagrams which normally arrange points in regular hexagonal grids. By measuring distances with an manhattan metric whose main axis is adjusted locally to follow the chosen direction field, the centroidal diagram can be adapted to place tiles in curving square grids instead. Computing the centroidal voronoi diagram is made possible by leveraging the z-buffer algorithm available in many graphics cards. 1

A new paradigm for rigid body simulation is presented and analyzed. 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 threedimensional 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. The extremely small integration steps inherent to traditional simulation techniques are avoided. Contact points are synchronized at the end of each frame. Resolving contacts with friction is known to be a difficult problem. Analytic force calculation can have ambiguous or non-existing solutions. Purely impulsive techniques avoid these ambiguous cases, but still require an excessive and computationally expensive number of updates in the case of

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...

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...

Layout and packing are NP-hard 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 NP-hard, 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...

We present exact algorithms for finding a solution to the two-dimensional 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...

Designers often need to decompose a product into functioning parts during the product design stage. This decomposition is critical for product development, as it determines the geometric configuration of parts, and has direct impact on product cost. Most decomposition decisions are based primarily upon end-user requirements instead of product manufacturability. The resulting parts can be expensive to manufacture or are sometimes impossible to make. This thesis presents a manufacturability-driven approach which can help designers decompose bent sheet metal products into manufacturable parts. The decomposition approach presented in this thesis takes the geometric description of an initial product design, analyzes its manufacturability, and decomposes the product into manufacturable parts. The decomposition continues until all decomposed parts are manufacturable. Near-optimal solutions are generated based on some primary concerns of design for manufacture (DFM) and design for assembly (DFA). Designers can then examine the decomposition results and decide whether they meet end-user requirements. Cutting, bending, and assembly processes are considered as the major manufacturing

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...