Packing Optimisation Using a Rubberband Analogy

2001 ◽  
Author(s):  
Georges M. Fadel ◽  
Avijit Sinha ◽  
Todd McKee
Keyword(s):  
Convex Hull ◽  
Relative Position ◽  
Dimensional Space ◽  
Packing Problem ◽  
Rubber Band ◽  
Single Component ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Optimal Packing ◽  
3 Dimensional

Abstract Packing is a topic of interest in many fields. During our research on the underhood-packing problem, we observed that to bring components together — without switching their relative position — one could use the analogy of a rubber object stretched around the artifacts. In two-dimensional space, that object is a rubber band, and in three-dimensions, it is a balloon. Using this analogy, the convex hull can be used to determine the direction of forces applied to a single component, and a motion can result from the application of such forces. The objects can then be moved until contact occurs, at which point the forces become moments, and the objects can rotate with respect to each other. This technique can guarantee locally optimal packing, and displays a very intuitive behavior that might lead to further advances in optimization. This paper introduces the methodology for optimizing the packing of 2-dimensional geometric entities (polygons) in a plane and of 3-dimensional objects in space using the Rubber band Analogy.

Digital Modeling and Movement Resolution of Rubber Band Analogy for 2D Packing Problem

2012 ◽  
Vol 220-223 ◽  
pp. 2466-2470 ◽  
Author(s):  
Jun Yan Ma ◽  
Xiao Ping Liao ◽  
Juan Lu ◽  
Hong Yao
Keyword(s):  
Convex Hull ◽  
Extreme Points ◽  
Packing Problem ◽  
Rubber Band ◽  
Scanning Method ◽  
Resultant Vector ◽  
Digital Modeling ◽  
Volume Decrease ◽  
Modeling Algorithm

Packing problem is how to arrange the components in available spaces to make the layout compact. This paper adopts a digital modeling algorithm to establish a novel rubber band convex hull model to solve this problem. A ray scanning method analogy QuickHull algorithm is presented to get extreme points of rubber band convex hull. A plural vector expression approach is adopted to movement resolution,which calculate the resultant vector to translate, rotate and slide the subbody to make the volume decrease. An experiment proved this digital modeling algorithm effective.

Some Helical Structures Generated by Reflexions

1985 ◽  
Vol 38 (3) ◽  
pp. 299 ◽  
Author(s):  
AC Hurley
Keyword(s):  
Dimensional Space ◽  
Helical Structure ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Helical Structures ◽  
Higher Dimensional ◽  
Translation Component ◽  
Explicit Formulae ◽  
Different Dimensions ◽  
Early Discussion

There has recently been a revival of interest in the helical structure built up as a column of face-sharing tetrahedra, because of possible applications in structural crystallography (Nelson 1983). This structure and its analogues in spaces of different dimensions are investigated here. It is shown that the only crystallographic cases are the structures in one- and two-dimensional space. For three and higher dimensional space the structures are all non-crystallographic. For the physically important case of three dimensions, this result is implicit in an early discussion by Coxeter (1969). Results obtained here include explicit formulae for the positions of all vertices of the simplexes for dimensions n = 1-4 and a demonstration that, for arbitrary n, the ratio of the translation component of the screw to the edge of the simplex is {6/ n(n+ I)(n+ 2)}1/2

scholarly journals Limitations of LEED-Structure Determination of Reconstructed Crystal Surfaces

1982 ◽  
Vol 37 (10) ◽  
pp. 1103-1118 ◽  
Author(s):  
H. Jagodzinski
Keyword(s):  
Structure Determination ◽  
Crystal Surface ◽  
Dimensional Space ◽  
Point Group ◽  
Two Dimensional ◽  
Crystal Surfaces ◽  
Difficult Situation ◽  
3 Dimensional ◽  
Lattice Constants

Although a strictly ordered reconstructed crystal surface may have the two-dimensional symmetry prescribed by the bulk (two-dimensional subsymmetry of the 3-dimensional space group), it belongs more frequently to a subgroup. In the LEED-pattern a lower symmetry of the translation group can easily be detected on account of the superstructure reflections observed. In this case anti-phase domains can hardly be avoided. An even more difficult situation arises if the symmetry of the point group is violated, although the symmetry of the diffraction pattern is not altered at the end of the reconstruction. Twin domains without changes of lattice constants have to be taken into account.Dynamical scattering of anti-phase domains is calculated by applying the "Cluster embedded chain"-method for various distributions of domains. It is shown that the incoherent superposition of scattering amplitudes, normally applied in LEED-calculations may lead to serious errors in structure determination. This effect is even more pronounced for twin domains. As an example we discuss the reconstruction of the (001)-surface of Si and Ge, and show that the wellknown (2 X 1)-structure is an averaged structure only, consisting of anti-phase domains with the (4 x 2)-structure. It may be concluded therefrom that none of the present model structures of this reconstruction agrees with this experimental observation.

Estimating Three-Dimensional Space from Multiple Two-Dimensional Views

1993 ◽  
Vol 2 (1) ◽  
pp. 44-53 ◽  
Author(s):  
Kristinn R. Thorisson
Keyword(s):  
Visual Search ◽  
Eye Movement ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Three Dimensions ◽  
Limiting Factor ◽  
Single Subject ◽  
Search Performance ◽  
Two Dimensional

The most common visual feedback technique in teleoperation is in the form of monoscopic video displays. As robotic autonomy increases and the human operator takes on the role of a supervisor, three-dimensional information is effectively presented by multiple, televised, two-dimensional (2-D) projections showing the same scene from different angles. To analyze how people go about using such segmented information for estimations about three-dimensional (3-D) space, 18 subjects were asked to determine the position of a stationary pointer in space; eye movements and reaction times (RTs) were recorded during a period when either two or three 2-D views were presented simultaneously, each showing the same scene from a different angle. The results revealed that subjects estimated 3-D space by using a simple algorithm of feature search. Eye movement analysis supported the conclusion that people can efficiently use multiple 2-D projections to make estimations about 3-D space without reconstructing the scene mentally in three dimensions. The major limiting factor on RT in such situations is the subjects' visual search performance, giving in this experiment a mean of 2270 msec (SD = 468; N = 18). This conclusion was supported by predictions of the Model Human Processor (Card, Moran, & Newell, 1983), which predicted a mean RT of 1820 msec given the general eye movement patterns observed. Single-subject analysis of the experimental data suggested further that in some cases people may base their judgments on a more elaborate 3-D mental model reconstructed from the available 2-D views. In such situations, RTs and visual search patterns closely resemble those found in the mental rotation paradigm (Just & Carpenter, 1976), giving RTs in the range of 5-10 sec.

scholarly journals An algorithm of packing congruent circles in a multiply connected set with non-Euclidean metrics

2016 ◽  
pp. 177-188
Author(s):  
А.Л. Казаков ◽  
А.А. Лемперт ◽  
Г.Л. Нгуен
Keyword(s):  
Numerical Experiments ◽  
Geometric Approach ◽  
Packing Problem ◽  
Two Dimensional ◽  
Optimal Packing ◽  
Multiply Connected ◽  
Bounded Set ◽  
Connected Set ◽  
Euclidean Metrics ◽  
Special Algorithm

Рассматривается задача об упаковке конгруэнтных кругов в ограниченное множество (контейнер) в двумерном метрическом пространстве: требуется найти такое расположение кругов в контейнере, при котором они заполнят как можно большую долю последнего. В случае, когда пространство является евклидовым, эта задача достаточно хорошо изучена, однако существует ряд прикладных задач, в частности в области инфраструктурной логистики, которые приводят нас к необходимости использовать специальные неевклидовые метрики. Исследованию таких задач и посвящена данная работа, причем рассматриваются как односвязные, так и многосвязные контейнеры. Разработан и программно реализован алгоритм численного решения указанной задачи, основанный на оптико-геометрическом подходе. Приведены результаты вычислительного эксперимента. The problem of optimal packing of congruent circles in a bounded set (a container) in a two-dimensional metric space is considered. It is required to find an arrangement of circles in the container such that these circles occupy the largest area of the container as possible. In the case when the space is Euclidean, this problem is well known, but the case of non-Euclidean metrics is studied much worse. However, there are some applied problems leading us to the use of special non-Euclidean metrics. For example, such a situation appears in the infrastructure logistics. Here we consider the optimal packing problem in the case when the container is simply or multiply connected. A special algorithm based on the optical-geometric approach is proposed and implemented. The results of numerical experiments are discussed.

Plane groups – from basic to advanced crystallographic concepts

2003 ◽  
Vol 218 (9) ◽  
Author(s):  
K. Stróż
Keyword(s):  
Packing Density ◽  
Circle Packing ◽  
Dimensional Space ◽  
Two Dimensional ◽  
Plane Symmetry ◽  
Contour Map ◽  
3 Dimensional ◽  
Complex Approach ◽  
Diffraction Patterns

AbstractThe plane groups are rarely discussed in the crystallography courses and the didactic role of the plane groups for teaching symmetry is rather underestimated. Most crystallographic concepts known from 3-dimensional space group descriptions concern also these 2-dimensional groups and can be easier illustrated. Symmetry of atomic layers as well as symmetry of mosaics, lattice designs, symmetry of electron diffraction patterns or any pattern with two-dimensional periodicity can be characterised on a unique basis. Such a pattern can be generated by decorating the points of a crystallographic orbit by different graphical objects. The orbits are visualised as “crystallographic mosaics” (by connection of the closest points of the orbit), they can be characterised by Shubnikov or Laves nets and there always exists a number (circle packing density) that is related to a given orbit. The paper and a didactic computer program presented in it give some ideas and framework for the plane symmetry experiments: from finding the isometries on the generated patterns to identifying the non-characteristic orbits. It is also shown that the contour map of a circle packing density is a good frame for locating symmetry elements, asymmetric units or points with higher plane symmetry into a unit cell. The applied complex approach to the orbit characterisation will narrow the gap existing between concepts used by practising and theoretical crystallographers and mathematicians interested in tessellations.

scholarly journals Three-dimensional space: locomotory style explains memory differences in rats and hummingbirds

2014 ◽  
Vol 281 (1784) ◽  
pp. 20140301 ◽  
Author(s):  
I. Nuri Flores-Abreu ◽  
T. Andrew Hurly ◽  
James A. Ainge ◽  
Susan D. Healy
Keyword(s):  
Horizontal Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Vertical Component ◽  
Horizontal Component ◽  
Vertical Dimension ◽  
Three Dimensions ◽  
Two Dimensional ◽  
With Memory ◽  
Three Dimensional Space

While most animals live in a three-dimensional world, they move through it to different extents depending on their mode of locomotion: terrestrial animals move vertically less than do swimming and flying animals. As nearly everything we know about how animals learn and remember locations in space comes from two-dimensional experiments in the horizontal plane, here we determined whether the use of three-dimensional space by a terrestrial and a flying animal was correlated with memory for a rewarded location. In the cubic mazes in which we trained and tested rats and hummingbirds, rats moved more vertically than horizontally, whereas hummingbirds moved equally in the three dimensions. Consistent with their movement preferences, rats were more accurate in relocating the horizontal component of a rewarded location than they were in the vertical component. Hummingbirds, however, were more accurate in the vertical dimension than they were in the horizontal, a result that cannot be explained by their use of space. Either as a result of evolution or ontogeny, it appears that birds and rats prioritize horizontal versus vertical components differently when they remember three-dimensional space.

scholarly journals Determination of load distribution in a given medium according to the values of the loads at certain points

2021 ◽  
pp. 167-175
Author(s):  
Oleksandr Mostovenko ◽  
Serhii Kovalov ◽  
Svitlana Botvinovska
Keyword(s):  
Boundary Conditions ◽  
Load Distribution ◽  
Numerical Solutions ◽  
Dimensional Space ◽  
Geometric Model ◽  
Three Dimensional ◽  
The Body ◽  
Three Dimensions ◽  
Two Dimensional ◽  
One Dimensional

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2]. In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7]. From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes. Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions. If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid. If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes. The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells. As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.

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