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Journal articles on the topic 'Rectangle'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 January 2023

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1

Kwon, Bo-Hyun, and Jung Hoon Lee. "Properties of Casson–Gordon’s rectangle condition." Journal of Knot Theory and Its Ramifications 29, no. 12 (October 2020): 2050083. http://dx.doi.org/10.1142/s0218216520500832.

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For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

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2

Huang, Eric, and Richard Korf. "Optimal Rectangle Packing on Non-Square Benchmarks." Proceedings of the AAAI Conference on Artificial Intelligence 24, no. 1 (July 3, 2010): 83–88. http://dx.doi.org/10.1609/aaai.v24i1.7538.

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The rectangle packing problem consists of finding an enclosing rectangle of smallest area that can contain a given set of rectangles without overlap. We propose two new benchmarks, one where the orientation of the rectangles is fixed and one where it is free, that include rectangles of various aspect ratios. The new benchmarks avoid certain properties of easy instances, which we identify as instances where rectangles have dimensions in common or where a few rectangles occupy most of the area. Our benchmarks are much more difficult for the previous state-of-the-art solver, requiring orders of magnitude more time, compared to similar-sized instances from a popular benchmark consisting only of squares. On the new benchmarks, we improve upon the previous strategy used to handle dominance conditions, we define a variable order over non-square rectangles that generalizes previous strategies, and we present a way to adjust the sizes of intervals of values for each rectangle's x-coordinates. Using these techniques together, we can solve the new oriented benchmark about 500 times faster, and the new unoriented benchmark about 40 times faster than the previous state-of-the-art.

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3

Ellard, Richard, and Des MacHale. "Packing a rectangle with m x (m + 1) rectangles." Mathematical Gazette 100, no. 547 (March 2016): 34–47. http://dx.doi.org/10.1017/mag.2016.6.

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We consider the packing of rectangles of dimension m x (m + 1) — where m is a natural number — into a larger rectangle. More specifically, we consider the following problem: What is the smallest area of a rectangle into which rectangles of dimensions 1 x 2, 2 x 3, 3 x 4,…, n x (n + 1) will fit without overlap? Unlike the corresponding problem for squares of areas 12, 22, 32, …, n2(see [1]), where there is no known non-trivial example of an exact fit into a rectangle, in many cases we can achieve an exact fit for our set of m x (m + 1) rectangles. Intuitively, this is because each m x (m + 1) rectangle has two possible orientations, which considerably increases the chances of an exact fit. As in [1], we make the (possibly unnecessary) assumption that the sides of each m x (m + 1) rectangle are parallel to the sides of the bounding rectangle, whose dimensions are integral. For any given n, we consider two solutions to our problem to be distinct only if the bounding rectangles have different dimensions (but equal area).

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4

NAGAMOCHI, HIROSHI. "PACKING SOFT RECTANGLES." International Journal of Foundations of Computer Science 17, no. 05 (October 2006): 1165–78. http://dx.doi.org/10.1142/s0129054106004327.

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Let R be a rectangle with given area a(R), height h(R) and width w(R), and r1, r2, …, rn be n soft rectangles, where we mean by a soft rectangle a rectangle r whose area a(r) is prescribed but whose aspect ratio ρ(r) is allowed to be changed. In this paper, we consider the problem of packing n soft rectangles r1, r2, …, rn into R. We prove that, if a(R) ≥ Σ1≤i≤n a(ri) + 0.10103amax and amax ≤ 3( min {h(R), w(R)})2 hold for a amax = max 1≤i≤n a(ri), then these n soft rectangles can be packed inside R so that the apect ratio of each rectangle ri is at most 3.

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5

Savic, Aleksandar, Jozef Kratica, and Vladimir Filipovic. "A new nonlinear model for the two-dimensional rectangle packing problem." Publications de l'Institut Math?matique (Belgrade) 93, no. 107 (2013): 95–107. http://dx.doi.org/10.2298/pim1307095s.

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This paper deals with the rectangle packing problem, of filling a big rectangle with smaller rectangles, while the rectangle dimensions are real numbers. A new nonlinear programming formulation is presented and the validity of the formulation is proved. In addition, two cases of the problem are presented, with and without rotation of smaller rectangles by 90?. The mixed integer piecewise linear formulation derived from the model is given, but with a simple form of the objective function.

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6

KIM, SANG-SUB, SANG WON BAE, and HEE-KAP AHN. "COVERING A POINT SET BY TWO DISJOINT RECTANGLES." International Journal of Computational Geometry & Applications 21, no. 03 (June 2011): 313–30. http://dx.doi.org/10.1142/s0218195911003676.

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Given a set S of n points in the plane, the disjoint two-rectangle covering problem is to find a pair of disjoint rectangles such that their union contains S and the area of the larger rectangle is minimized. In this paper we consider two variants of this optimization problem: (1) the rectangles are allowed to be reoriented freely while restricting them to be parallel to each other, and (2) one rectangle is restricted to be axis-parallel but the other rectangle is allowed to be reoriented freely. For both of the problems, we present O(n2 log n)-time algorithms using O(n) space.

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7

BIRD, RICHARD S. "Building a consensus: A rectangle covering problem." Journal of Functional Programming 21, no. 2 (January 5, 2011): 119–28. http://dx.doi.org/10.1017/s0956796810000316.

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The other day, over a very pleasant lunch in the restaurant of Oxford's recently renovated Ashmolean Museum, Oege de Moor gave me a problem about rectangles. The problem is explained more fully later, but roughly speaking one is given a finite set of rectangles RS and a rectangle R completely covered by RS. The task is to construct a single rectangle covering R among the elements of a larger set of rectangles associated with RS, called the saturation of RS. The saturation of RS is the closure of RS under so-called consensus operations, a term coined in (Quine, 1959), in which two rectangles are combined in two distinct ways to form new rectangles. The rectangle problem is a simplified version of containment-checking, a crucial component in a type inference algorithm for Datalog programs (Schäfer & de Moor, 2010). 19 In the Schäfer-de Moor algorithm the problem is generalised to cubes in n-space rather than rectangles in two-space, the components of each cube are given by propositional formulae rather than by intervals on the real line, and certain equality and inhabitation constraints are taken into account. Oege felt that the central proof, Lemma 15 in (Schäfer & de Moor, 2010), deserved to be simplified so he posed the rectangle problem as a special case. This pearl was composed in response to the challenge.

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8

Alon, Noga, and Daniel J. Kleitman. "Partitioning a rectangle into small perimeter rectangles." Discrete Mathematics 103, no. 2 (May 1992): 111–19. http://dx.doi.org/10.1016/0012-365x(92)90261-d.

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9

Joós, Antal. "On packing of rectangles in a rectangle." Discrete Mathematics 341, no. 9 (September 2018): 2544–52. http://dx.doi.org/10.1016/j.disc.2018.06.007.

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10

Huang, Eric, and Richard Korf. "Optimal Packing of High-Precision Rectangles." Proceedings of the International Symposium on Combinatorial Search 2, no. 1 (August 19, 2021): 195–96. http://dx.doi.org/10.1609/socs.v2i1.18211.

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The rectangle-packing problem consists of finding an enclosing rectangle of smallest area that can contain a given set of rectangles without overlap. Our new benchmark includes rectangles of successively higher precision, a problem for the previous state-of-the-art, which enumerates all locations for placing rectangles. We instead limit these locations and bounding box dimensions to the set of subset sums of the rectangles' dimensions, allowing us to test 4,500 times fewer bounding boxes and solve N=9 over two orders of magnitude faster. Finally, on the open problem of the feasibility of packing a specific infinite series of rectangles into the unit square, we pack the first 50,000 such rectangles and conjecture that the entire infinite series can fit.

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11

Zavagno, D. "Some New Effects: Phenomenal Glare, Luminous ‘Mist’ and Dark ‘Smoke’." Perception 26, no. 1_suppl (August 1997): 58. http://dx.doi.org/10.1068/v970247.

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The impression of glare is caused by a very intense light source. However, here I show that this impression can also be generated with normal light intensities. The strength of the effect depends on the number of elements used to produce it. The elements are 2 cm × 5 cm rectangles. A single horizontal achromatic rectangle is first used on a homogeneous white or black background. From left to right, the brightness of the rectangle varies smoothly from black to white. The left part of the rectangle appears to progressively bend toward the background when the background is black, while the rectangle appears straight and to fade into an apparent white mist near its right side when the background is white. When the background is black, two horizontal rectangles, mirror-shaded from black to white, so that their black ends face each other with a 2 cm gap between them, appear either to bend toward the background or to be straight and to fade into a sort of dark ‘smoke’. When the background is white with the left rectangle varying in brightness from black to white and the right one from white to black, the rectangles look straight with a sort of white glare appearing to come out from the gap. The black ‘smoke’ and the white glare look more compelling when there are four rectangles forming a cross with a central square gap. It can be argued that this and the neon spreading effect are unrelated. Instead, psychophysical experiments suggest that the glare and smoke effects depend on a relation between the grey scale gradient and the background brightness.

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12

White, Peter A. "Visual Impressions of Interactions between Objects When the Causal Object Does Not Move." Perception 34, no. 4 (April 2005): 491–500. http://dx.doi.org/10.1068/p3263.

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Stimuli were presented that consisted of a stationary row of black-bordered white rectangles. As observers watched, each rectangle in turn from left to right changed from white to black. The final rectangle did not change colour but moved off from left to right. The sequential colour change suggested motion from left to right, and observers reliably reported a visual impression that this illusory motion kicked or bumped the last rectangle, thereby making it move. The impression was stronger when the sequential colour change was faster, but was not significantly affected by the number of the rectangles in the row (ranging from 2 to 8). These results support the conclusion that neither contact nor motion of a causal object is necessary for a visual impression of causality to occur.

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13

Recuero, A., M. Álvarez, and O. Río. "Realización de un grafo en recintos rectangulares sobre una planta definida." Informes de la Construcción 47, no. 437 (June 30, 1995): 63–85. http://dx.doi.org/10.3989/ic.1995.v47.i437.1074.

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14

Jia, Jie, Yong Jun Yang, Yi Ming Hou, Xiang Yang Zhang, and He Huang. "Adaboost Classification-Based Object Tracking Method for Sequence Images." Applied Mechanics and Materials 44-47 (December 2010): 3902–6. http://dx.doi.org/10.4028/www.scientific.net/amm.44-47.3902.

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An object tracking framework based on adaboost and Mean-Shift for image sequence was proposed in the manuscript. The object rectangle and scene rectangle in the initial image of the sequence were drawn and then, labeled the pixel data in the two rectangles with 1 and 0. Trained the adaboost classifier by the pixel data and the corresponding labels. The obtained classifier was improved to be a 5 class classifier and employed to classify the data in the same scene region of next image. The confidence map including 5 values was got. The Mean-Shift algorithm is performed in the confidence map area to get the final object position. The rectangles of object and background were moved to the new position. The object rectangle was zoomed by 5 percent to adapt the object scale changing. The process including drawing rectangle, training, classification, orientation and zooming would be repeated until the end of the image sequence. The experiments result showed that the proposed algorithm is efficient for nonrigid object orientation in the dynamic scene.

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15

CHEN, DUANBING, and WENQI HUANG. "A NEW HEURISTIC ALGORITHM FOR CONSTRAINED RECTANGLE-PACKING PROBLEM." Asia-Pacific Journal of Operational Research 24, no. 04 (August 2007): 463–78. http://dx.doi.org/10.1142/s0217595907001334.

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The constrained rectangle-packing problem is the problem of packing a subset of rectangles into a larger rectangular container, with the objective of maximizing the layout value. It has many industrial applications such as shipping, wood and glass cutting, etc. Many algorithms have been proposed to solve it, for example, simulated annealing, genetic algorithm and other heuristic algorithms. In this paper a new heuristic algorithm is proposed based on two strategies: the rectangle selecting strategy and the rectangle packing strategy. We have applied the algorithm to 21 smaller, 630 larger and other zero-waste instances. The computational results demonstrate that the integrated performance of the algorithm is rather satisfying and the algorithm developed is fairly efficient for solving the constrained rectangle-packing problem.

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16

Snay, R. A., H. C. Neugebauer, and W. H. Prescott. "Horizontal deformation associated with the Loma Prieta earthquake." Bulletin of the Seismological Society of America 81, no. 5 (October 1, 1991): 1647–59. http://dx.doi.org/10.1785/bssa0810051647.

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Abstract Co-seismic horizontal displacements for the 1989 Loma Prieta earthquake were derived from preseismic triangulation/trilateration observations and post-seismic GPS observations. As part of this process, the empirical model entitled TDP-H91 was applied to “correct” the preseismic measurements for the crustal motion that occurred during the seven decades spanned by these data. These newly derived displacements were combined with previously documented geodetic results to generate a dislocation model for the earthquake. Our preferred model consists of a vertically segmented rupture surface represented by two rectangles that share a common edge at a depth of 9 km. The upper rectangle dips 90° and the lower rectangle dips 70°SW. Via a trial-and-error technique, the following estimates were found for the remaining parameters: strike = 134.4 ± 0.7°, fault length = 32.4 ± 0.7 km, upper depth = 4.8 ± 0.1 km, lower depth = 15.1 ± 0.3 km, right-lateral strike slip = 1.86 ± 0.06 m for the upper rectangle and 1.96 ± 0.13 m for the lower rectangle, and thrusting dip slip = 1.06 ± 0.06 m for the upper rectangle and 2.30 ± 0.18 m for the lower rectangle.

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17

HÄGGKVIST, ROLAND, and ANDERS JOHANSSON. "Orthogonal Latin Rectangles." Combinatorics, Probability and Computing 17, no. 4 (July 2008): 519–36. http://dx.doi.org/10.1017/s0963548307008590.

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We use a greedy probabilistic method to prove that, for every ε > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1 − ε)n. That is, we show the existence of a second Latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.

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18

Zaidi, Abdelhamid. "Mathematical Methods for IoT-Based Annotating Object Datasets with Bounding Boxes." Mathematical Problems in Engineering 2022 (August 23, 2022): 1–16. http://dx.doi.org/10.1155/2022/3001939.

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Object datasets used in the construction of object detectors are typically annotated with horizontal or oriented bounding rectangles for IoT-based. The optimality of an annotation is obtained by fulfilling two conditions: (i) the rectangle covers the whole object and (ii) the area of the rectangle is minimal. Building a large-scale object dataset requires annotators with equal manual dexterity to carry out this tedious work. When an object is horizontal for IoT-based, it is easy for the annotator to reach the optimal bounding box within a reasonable time. However, if the object is oriented, the annotator needs additional time to decide whether the object will be annotated with a horizontal rectangle or an oriented rectangle for IoT-based. Moreover, in both cases, the final decision is not based on any objective argument, and the annotation is generally not optimal. In this study, we propose a new method of annotation by rectangles for IoT-based, called robust semi-automatic annotation, which combines speed and robustness. Our method has two phases. The first phase consists in inviting the annotator to click on the most relevant points located on the contour of the object. The outputs of the first phase are used by an algorithm to determine a rectangle enclosing these points. To carry out the second phase, we develop an algorithm called RANGE-MBR, which determines, from the selected points on the contour of the object, a rectangle enclosing these points in a linear time. The rectangle returned by RANGE-MBR always satisfies optimality condition (i). We prove that the optimality condition (ii) is always satisfied for objects with isotropic shapes. For objects with anisotropic shapes, we study the optimality condition (ii) by simulations. We show that the rectangle returned by RANGE-MBR is quasi-optimal for the condition (ii) and that its performance increases with dilated objects, which is the case for most of the objects appearing on images collected by aerial photography.

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19

Erb, Christopher D., Jeff Moher, Joo-Hyun Song, and David M. Sobel. "Numerical cognition in action: Reaching behavior reveals numerical distance effects in 5- to 6-year-olds." Journal of Numerical Cognition 4, no. 2 (September 7, 2018): 286–96. http://dx.doi.org/10.5964/jnc.v4i2.122.

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This study investigates how children’s numerical cognition is reflected in their unfolding actions. Five- and 6-year-olds (N = 34) completed a numerical comparison task by reaching to touch one of three rectangles arranged horizontally on a digital display. A number from 1 to 9 appeared in the center rectangle on each trial. Participants were instructed to touch the left rectangle for numbers 1-4, the center rectangle for 5, and the right rectangle for 6-9. Reach trajectories were more curved toward the center rectangle for numbers closer to 5 (e.g., 4) than numbers further from 5 (e.g., 1). This finding indicates that a tight coupling exists between numerical and spatial information in children’s cognition and action as early as the preschool years. In addition to shedding new light on the spatial representation of numbers during childhood, our results highlight the promise of incorporating measures of manual dynamics into developmental research.

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20

STEWART, ROBERT, and HONG ZHANG. "A NOTE CONCERNING THE DISTANCES OF UNIFORMLY DISTRIBUTED POINTS FROM THE CENTRE OF A RECTANGLE." Bulletin of the Australian Mathematical Society 87, no. 1 (June 7, 2012): 115–19. http://dx.doi.org/10.1017/s0004972712000421.

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AbstractGiven a rectangle containing uniformly distributed random points, how far are the points from the rectangle’s centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.

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21

Ren, Hong E., Mian Liu, and Meng Zhu. "Recognition Analysis of Wood Flour Mesh Number Based on External Rectangle Fitting Algorithm." Applied Mechanics and Materials 496-500 (January 2014): 1995–98. http://dx.doi.org/10.4028/www.scientific.net/amm.496-500.1995.

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To overcome disadvantages of traditional detection methods of wood flour mesh number, a mesh number recognition algorithm based on external rectangle fitting and morphological characteristics has been studied. It makes use of minimum external rectangle with the boundary points obtained by the preprocessing of microscopic images. The external rectangles length is calculated when the area is the smallest. The experimental results demonstrate that the proposed algorithm has a good fitting accuracy and meets producing demands.

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22

Wilson, Patricia S., and Verna M. Adams. "A Dynamic Way to Teach Angle and Angle Measure." Arithmetic Teacher 39, no. 5 (January 1992): 6–13. http://dx.doi.org/10.5951/at.39.5.0006.

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23

Huang, Eric, and Richard Korf. "Optimal Packing of High-Precision Rectangles." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (August 4, 2011): 42–47. http://dx.doi.org/10.1609/aaai.v25i1.7814.

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The rectangle-packing problem consists of ﬁnding an enclosing rectangle of smallest area that can contain a given set of rectangles without overlap. Our new benchmark includes rectangles of successively higher precision, challenging the previous state-of-the-art, which enumerates all locations for placing rectangles, as well as all bounding box widths and heights up to the optimal box. We instead limit the rectangles’ coordinates and bounding box dimensions to the set of subset sums of the rectangles’ dimensions. We also dynamically prune values by learning from infeasible subtrees and constrain the problem by replacing rectangles and empty space with larger rectangles. Compared to the previous state-of-the-art, we test 4,500 times fewer bounding boxes on the high-precision benchmark and solve N =9 over two orders of magnitude faster. We also present all optimal solutions up to N =15, which requires 39 bits of precision to solve. Finally, on the open problem of whether or not one can pack a particular inﬁnite series of rectangles into the unit square, we pack the ﬁrst 50,000 such rectangles witha greedy heuristic and conjecture that the entire inﬁnite series can ﬁt..

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24

Yee, Wee L. "Three-Dimensional Versus Rectangular Sticky Yellow Traps for Western Cherry Fruit Fly (Diptera: Tephritidae)." Journal of Economic Entomology 112, no. 4 (April 16, 2019): 1780–88. http://dx.doi.org/10.1093/jee/toz092.

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Abstract The most effective traps tested against western cherry fruit fly, Rhagoletis indifferens Curran, have been the Yellow Sticky Strip (YSS) rectangle made of styrene and the three-dimensional yellow Rebell cross made of polypropylene. However, three-dimensional YSS styrene traps have never been tested against this or any other fruit fly. The main objectives of this study were to determine the efficacies of 1) YSS cross, Rebell cross, YSS cylinder, and YSS rectangle traps, 2) Rebell cross versus Rebell rectangle traps, and 3) YSS tent versus YSS rectangle traps for R. indifferens. For 1), the YSS cross caught more flies than the Rebell cross of equivalent surface area and more than a smaller YSS cylinder, but not any more than a YSS rectangle of similar surface area as the YSS cross. For 2), a Rebell cross caught more flies than a rectangle of equivalent surface area made of Rebell panels. For 3), YSS tent and YSS rectangle traps of equivalent surface area did not differ in fly captures. Results suggest that the YSS cross was more effective than the Rebell cross due to its color and that when trap color is highly attractive, three-dimensional shape may be unimportant, whereas it could be when trap color is less attractive. A new trap modeled after the YSS cross, compact but with high trap surface area to increase fly captures, could be an effective option or addition to rectangles for monitoring R. indifferens.

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25

Oh, Eunjin, and Hee-Kap Ahn. "Finding pairwise intersections of rectangles in a query rectangle." Computational Geometry 85 (December 2019): 101576. http://dx.doi.org/10.1016/j.comgeo.2019.101576.

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26

Huang, E., and R. E. Korf. "Optimal Rectangle Packing: An Absolute Placement Approach." Journal of Artificial Intelligence Research 46 (January 23, 2013): 47–87. http://dx.doi.org/10.1613/jair.3735.

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We consider the problem of finding all enclosing rectangles of minimum area that can contain a given set of rectangles without overlap. Our rectangle packer chooses the x-coordinates of all the rectangles before any of the y-coordinates. We then transform the problem into a perfect-packing problem with no empty space by adding additional rectangles. To determine the y-coordinates, we branch on the different rectangles that can be placed in each empty position. Our packer allows us to extend the known solutions for a consecutive-square benchmark from 27 to 32 squares. We also introduce three new benchmarks, avoiding properties that make a benchmark easy, such as rectangles with shared dimensions. Our third benchmark consists of rectangles of increasingly high precision. To pack them efficiently, we limit the rectangles' coordinates and the bounding box dimensions to the set of subset sums of the rectangles' dimensions. Overall, our algorithms represent the current state-of-the-art for this problem, outperforming other algorithms by orders of magnitude, depending on the benchmark.

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27

DINITZ, YEFIM, MATTHEW J. KATZ, and ROI KRAKOVSKI. "GUARDING RECTANGULAR PARTITIONS." International Journal of Computational Geometry & Applications 19, no. 06 (December 2009): 579–94. http://dx.doi.org/10.1142/s0218195909003131.

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A rectangular partition is a partition of a rectangle into non-overlapping rectangles, such that no four rectangles meet at a common point. A vertex guard is a guard located at a vertex of the partition (i.e., at a corner of a rectangle); it guards the rectangles that meet at this vertex. An edge guard is a guard that patrols along an edge of the partition, and is thus equivalent to two adjacent vertex guards. We consider the problem of finding a minimum-cardinality guarding set for the rectangles of the partition. For vertex guards, we prove that guarding a given subset of the rectangles is NP-hard. For edge guards, we prove that guarding all rectangles, where guards are restricted to a given subset of the edges, is NP-hard. For both results we show a reduction from vertex cover in non-bipartite 3-connected cubic planar graphs of girth greater than three. For the second NP-hardness result, we obtain a graph-theoretic result which establishes a connection between the set of faces of a plane graph of vertex degree at most three and a vertex cover for this graph. More precisely, we prove that one can assign to each internal face a distinct vertex of the cover, which lies on the face's boundary. We show that the vertices of a rectangular partition can be colored red, green, or black, such that each rectangle has all three colors on its boundary. We conjecture that the above is also true for four colors. Finally, we obtain a worst-case upper bound on the number of edge guards that are sufficient for guarding rectangular partitions with some restrictions on their structure.

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28

Li, Ziqiang, Xianfeng Wang, Jiyang Tan, and Yishou Wang. "A Quasiphysical and Dynamic Adjustment Approach for Packing the Orthogonal Unequal Rectangles in a Circle with a Mass Balance: Satellite Payload Packing." Mathematical Problems in Engineering 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/657170.

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Packing orthogonal unequal rectangles in a circle with a mass balance (BCOURP) is a typical combinational optimization problem with the NP-hard nature. This paper proposes an effective quasiphysical and dynamic adjustment approach (QPDAA). Two embedded degree functions between two orthogonal rectangles and between an orthogonal rectangle and the container are defined, respectively, and the extruded potential energy function and extruded resultant force formula are constructed based on them. By an elimination of the extruded resultant force, the dynamic rectangle adjustment, and an iteration of the translation, the potential energy and static imbalance of the system can be quickly decreased to minima. The continuity and monotony of two embedded degree functions are proved to ensure the compactness of the optimal solution. Numerical experiments show that the proposed QPDAA is superior to existing approaches in performance.

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29

Sari, Siska Nurmala. "HUBUNGAN ANTARA HIMPUNAN KUBIK ASIKLIK DENGAN RECTANGLE." Jurnal Matematika UNAND 3, no. 1 (March 1, 2014): 53. http://dx.doi.org/10.25077/jmu.3.1.53-57.2014.

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Dalam artikel ini akan dipelajari hubungan antara himpunan kubik asiklik dengan rectangle. Diberikan suatu kubus dasar Q yang merupakan suatu hasil kaliberhingga dari interval-interval dasar I = [l, l +1] atau I = [l, l] untuk suatu l ∈ Z. suatuhimpunan kubik X adalah gabungan berhingga dari kubus-kubus dasar Q. Himpunankubik dengan bentuk X = [k 1 , l 1 ] × [k 2 , l 2] × · · · × [kn , ln] ⊂ Rn disebut rectangle, dimanaki , li adalah bilangan bulat dan ki ≤ li. Selanjutnya diperoleh bahwa sebarang rectangleX adalah asiklik, dengan kata lain Hk(X) isomorfik dengan Z jika k = 0, dan Hk(X)isomorfik dengan 0 jika k > 0.

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30

Virk, Amandeep K., and Kawaljeet Singh. "On Performance of Binary Flower Pollination Algorithm for Rectangular Packing Problem." Recent Advances in Computer Science and Communications 13, no. 1 (March 13, 2020): 22–34. http://dx.doi.org/10.2174/2213275911666181114143239.

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Background: Metaheuristic algorithms are optimization algorithms capable of finding near-optimal solutions for real world problems. Rectangle Packing Problem is a widely used industrial problem in which a number of small rectangles are placed into a large rectangular sheet to maximize the total area usage of the rectangular sheet. Metaheuristics have been widely used to solve the Rectangle Packing Problem. Objective: A recent metaheuristic approach, Binary Flower Pollination Algorithm, has been used to solve for rectangle packing optimization problem and its performance has been assessed. Methods: A heuristic placement strategy has been used for rectangle placement. Then, the Binary Flower Pollination Algorithm searches the optimal placement order and optimal layout. Results: Benchmark datasets have been used for experimentation to test the efficacy of Binary Flower Pollination Algorithm on the basis of utilization factor and number of bins used. The simulation results obtained show that the Binary Flower Pollination Algorithm outperforms in comparison to the other well-known algorithms. Conclusion: BFPA gave superior results and outperformed the existing state-of-the-art algorithms in many instances. Thus, the potential of a new nature based metaheuristic technique has been discovered.

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Jansen, Klaus, and Guochuan Zhang. "Maximizing the Total Profit of Rectangles Packed into a Rectangle." Algorithmica 47, no. 3 (March 2007): 323–42. http://dx.doi.org/10.1007/s00453-006-0194-5.

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Kong, T. Y., David M. Mount, and A. W. Roscoe. "The Decomposition of a Rectangle into Rectangles of Minimal Perimeter." SIAM Journal on Computing 17, no. 6 (December 1988): 1215–31. http://dx.doi.org/10.1137/0217077.

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Feldman, Jacob, and Whitman Richards. "Mapping the Mental Space of Rectangles." Perception 27, no. 10 (October 1998): 1191–202. http://dx.doi.org/10.1068/p271191.

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The cognitive structure of a shape space—the space of rectangles—is explored by a nonmetric scaling technique. Our experiment was designed to extract the major transformational paths or ‘modes’ that characterize the mental shape space. Earlier studies of rectangle similarities using multidimensional scaling have provided conflicting evidence about whether the coordinate system of the mental rectangle space is based on height and width or on area and shape (ie aspect ratio). Our study reveals shape to be the single dominant factor. We suspected that earlier evidence for a height – width parameterization might have been due to the presentation of rectangles upright in a pseudo-gravitational coordinate system (whereas our rectangles are randomly rotated). In a control experiment with upright (vertical or horizontal) rectangles, the heavy bias towards shape preservation was still the dominant mode. In addition, however, a secondary bias towards change of height or width emerged, exactly following the pattern expected from the biasing change in context. This finding established a concrete path by which context and frame can influence the way shape is represented. The relevance of these findings to the cognitive organization of more complex shape spaces is discussed.

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Suryaningrum, Christine Wulandari, Purwanto Purwanto, Subanji Subanji, Hery Susanto, Yoga Dwi Windy Kusuma Ningtyas, and Muhammad Irfan. "SEMIOTIC REASONING EMERGES IN CONSTRUCTING PROPERTIES OF A RECTANGLE: A STUDY OF ADVERSITY QUOTIENT." Journal on Mathematics Education 11, no. 1 (January 25, 2020): 95–110. http://dx.doi.org/10.22342/jme.11.1.9766.95-110.

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Semiotics is simply defined as the sign-using to represent a mathematical concept in a problem-solving. Semiotic reasoning of constructing concept is a process of drawing a conclusion based on object, representamen (sign), and interpretant. This paper aims to describe the phases of semiotic reasoning of elementary students in constructing the properties of a rectangle. The participants of the present qualitative study are three elementary students classified into three levels of Adversity Quotient (AQ): quitter/AQ low, champer/AQ medium, and climber/AQ high. The results show three participants identify object by observing objects around them. In creating sign stage, they made the same sign that was a rectangular image. However, in three last stages, namely interpret sign, find out properties of sign, and discover properties of a rectangle, they made different ways. The quitter found two characteristics of rectangular objects then derived it to be a rectangle’s properties. The champer found four characteristics of the objects then it was derived to be two properties of a rectangle. By contrast, Climber found six characteristics of the sign and derived all of these to be four properties of a rectangle. In addition, Climber could determine the properties of a rectangle correctly.

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ANZAI, SHINYA, JINHEE CHUN, RYOSEI KASAI, MATIAS KORMAN, and TAKESHI TOKUYAMA. "EFFECT OF CORNER INFORMATION IN SIMULTANEOUS PLACEMENT OF k RECTANGLES AND TABLEAUX." Discrete Mathematics, Algorithms and Applications 02, no. 04 (December 2010): 527–37. http://dx.doi.org/10.1142/s1793830910000863.

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We consider the optimization problem of finding k nonintersecting rectangles and tableaux in n × n pixel plane where each pixel has a real valued weight. We discuss existence of efficient algorithms if a corner point of each rectangle/tableau is specified.

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Liu, Hong Hai, and Xiang Hua Hou. "The Face Detection Research Based on Multi-Scale and Rectangle Feature." Applied Mechanics and Materials 198-199 (September 2012): 1383–88. http://dx.doi.org/10.4028/www.scientific.net/amm.198-199.1383.

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When extracting the face image features based on pixel distribution in face image, there always exist large amount of calculation and high dimensions of feature sector generated after feature extraction. This paper puts forward a feature extraction method based on prior knowledge of face and Haar feature. Firstly, the Haar feature expressions of face images are classified and the face features are decomposed into edge feature, line feature and center-surround feature, which are further concluded into the expressions of two rectangles, three rectangles and four rectangles. In addition, each rectangle varies in size. However, for face image combination, there exist too much redundancy and large calculation amount in this kind of expression. In order to solve the problem of large amount of calculation, the integral image is adopted to speed up the rectangle feature calculation. In addition, the thought based on classified trainer is adopted to reduce the redundancy expression. The results show that using face image of Haar feature expression can improve the speed and efficiency of recognition.

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Yildiz, Gizem Y., Bailey G. Evans, and Philippe A. Chouinard. "The Effects of Adding Pictorial Depth Cues to the Poggendorff Illusion." Vision 6, no. 3 (July 18, 2022): 44. http://dx.doi.org/10.3390/vision6030044.

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We tested if the misapplication of perceptual constancy mechanisms might explain the perceived misalignment of the oblique lines in the Poggendorff illusion. Specifically, whether these mechanisms might treat the rectangle in the middle portion of the Poggendorff stimulus as an occluder in front of one long line appearing on either side, causing an apparent decrease in the rectangle’s width and an apparent increase in the misalignment of the oblique lines. The study aimed to examine these possibilities by examining the effects of adding pictorial depth cues. In experiments 1 and 2, we presented a central rectangle composed of either large or small bricks to determine if this manipulation would change the perceived alignment of the oblique lines and the perceived width of the central rectangle, respectively. The experiments demonstrated no changes that would support a misapplication of perceptual constancy in driving the illusion, despite some evidence of perceptual size rescaling of the central rectangle. In experiment 3, we presented Poggendorff stimuli in front and at the back of a corridor background rich in texture and linear perspective depth cues to determine if adding these cues would affect the Poggendorff illusion. The central rectangle was physically large and small when presented in front and at the back of the corridor, respectively. The strength of the Poggendorff illusion varied as a function of the physical size of the central rectangle, and, contrary to our predictions, the addition of pictorial depth cues in both the central rectangle and the background decreased rather than increased the strength of the illusion. The implications of these results with regards to different theories are discussed. It could be the case that the illusion depends on both low-level and cognitive mechanisms and that deleterious effects occur on the former when the latter ascribes more certainty to the oblique lines being the same line receding into the distance.

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Miller. "MEASURABLE RECTANGLE." Real Analysis Exchange 19, no. 1 (1993): 194. http://dx.doi.org/10.2307/44153827.

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Ginat, David. "Rectangle cover." ACM Inroads 3, no. 3 (September 2012): 34–35. http://dx.doi.org/10.1145/2339055.2339068.

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Steele, Kenneth M., Mary Ellen Dello Stritto, and Willard L. Brigner. "A Looming-Recession Threshold." Perceptual and Motor Skills 82, no. 2 (April 1996): 604–6. http://dx.doi.org/10.2466/pms.1996.82.2.604.

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When the origin of magnification-minification of an outline rectangle had a horizontal locus which exceeded one-fourth of the rectangle's horizontal dimension, 16 observers of 21 reported apparent depth characteristic of looming and recession.

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Ridley, J. N. "Rectangles and spirals." Mathematical Gazette 105, no. 564 (October 13, 2021): 416–24. http://dx.doi.org/10.1017/mag.2021.108.

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Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

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Chen, Kaizhi, Jiahao Zhuang, Shangping Zhong, and Song Zheng. "Optimization Method for Guillotine Packing of Rectangular Items within an Irregular and Defective Slate." Mathematics 8, no. 11 (November 1, 2020): 1914. http://dx.doi.org/10.3390/math8111914.

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Research on the rectangle packing problems has mainly focused on rectangular raw material sheets without defects, while natural slate has irregular and defective characteristics, and the existing packing method adopts manual packing, which wastes material and is inefficient. In this work, we propose an effective packing optimization method for nature slate; to the best of our knowledge, this is the first attempt to solve the guillotine packing problem of rectangular items in a single irregular and defective slate. This method is modeled by the permutation model, uses the horizontal level (HL) heuristic proposed in this paper to obtain feasible solutions, and then applies the genetic algorithm to optimize the quality of solutions further. The HL heuristic is constructed on the basis of computational geometry and level packing. This heuristic aims to divide the irregular plate into multiple subplates horizontally, calculates the movable positions of the rectangle in the subplates, determines whether or not the rectangle can be packed in the movable positions through computational geometry, and fills the scraps appropriately. Theoretical analysis confirms that the rectangles obtained through the HL heuristic are inside the plate and do not overlap with the defects. In addition, the packed rectangles do not overlap each other and satisfy the guillotine constraint. Accordingly, the packing problem can be solved. Experiments on irregular slates with defects show that the slate utilization through our method is between 89% and 95%. This result is better than manual packing and can satisfy actual production requirements.

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Dalfó, C., M. A. Fiol, N. López, and A. Martínez-Pérez. "Decompositions of a rectangle into non-congruent rectangles of equal area." Discrete Mathematics 344, no. 6 (June 2021): 112389. http://dx.doi.org/10.1016/j.disc.2021.112389.

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ATALLAH, MIKHAIL J., and DANNY Z. CHEN. "ON CONNECTING RED AND BLUE RECTILINEAR POLYGONAL OBSTACLES WITH NONINTERSECTING MONOTONE RECTILINEAR PATHS." International Journal of Computational Geometry & Applications 11, no. 04 (August 2001): 373–400. http://dx.doi.org/10.1142/s0218195901000547.

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We present efficient algorithms for the problems of matching red and blue disjoint geometric obstacles in the plane and connecting the matched obstacle pairs with mutually nonintersecting paths that have useful geometric properties. We first consider matching n red and n blue disjoint rectilinear rectangles and connecting the n matched rectangle pairs with nonintersecting monotone rectilinear paths; each such path consists of O(n) segments and is not allowed to touch any rectangle other than the matched pair that it is linking. Based on a numbering scheme for certain geometric objects and on several useful geometric observations, we develop an O(n log n) time, O(n) space algorithm that produces a desired matching for rectilinear rectangles. If an explicit printing of all the n paths is required, then our algorithm takes O(n log n+λ) time and O(n) space, where λ is the total size of the desired output. We then extend these matching algorithms to other classes of red/blue polygonal obstacles. The numbering scheme also finds applications to other problems.

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Falcón, Raúl M., Víctor Álvarez, María Dolores Frau, Félix Gudiel, and María Belén Güemes. "Pseudococyclic Partial Hadamard Matrices over Latin Rectangles." Mathematics 9, no. 2 (January 6, 2021): 113. http://dx.doi.org/10.3390/math9020113.

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The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated.

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Garcia-Molla, Victor M., Pedro Alonso-Jordá, and Ricardo García-Laguía. "Parallel border tracking in binary images using GPUs." Journal of Supercomputing 78, no. 7 (January 19, 2022): 9817–39. http://dx.doi.org/10.1007/s11227-021-04260-y.

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AbstractBorder tracking in binary images is an important kernel for many applications. There are very efficient sequential algorithms, most notably, the algorithm proposed by Suzuki et al., which has been implemented for CPUs in well-known libraries. However, under some circumstances, it would be advantageous to perform the border tracking in GPUs as efficiently as possible. In this paper, we propose a parallel version of the Suzuki algorithm that is designed to be executed in GPUs and implemented in CUDA. The proposed algorithm is based on splitting the image into small rectangles. Then, a thread is launched for each rectangle, which tracks the borders in its associated rectangle. The final step is to perform the connection of the borders belonging to several rectangles. The parallel algorithm has been compared with a state-of-the-art sequential CPU version, using two different CPUs and two different GPUs for the evaluation. The computing times obtained show that in these experiments with the GPUs and CPUs that we had available, the proposed parallel algorithm running in the fastest GPU is more than 10 times faster than the sequential CPU routine running in the fastest CPU.

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Mark Keil, J. "Covering Orthogonal Polygons with Non-Piercing Rectangles." International Journal of Computational Geometry & Applications 07, no. 05 (October 1997): 473–84. http://dx.doi.org/10.1142/s0218195997000284.

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Given a simply connected orthogonal polygon P, a polynomial time algorithm is presented to cover the polygon with the minimum number of rectangles, under the restriction that if A and B are two overlapping rectangles in the cover then either A - B or B - A is connected. The algorithm runs in O(n log n + nm) time, where n is the number of vertices of P and m is the number of edges in the visibility graph of P that are either horizontal, vertical or form the diagonal of an empty rectangle.

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Crilly, Tony. "A Supergolden Rectangle." Mathematical Gazette 78, no. 483 (November 1994): 320. http://dx.doi.org/10.2307/3620208.

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Perkins, Annie, and Christy Pettis. "Running a Rectangle." Mathematics Teaching in the Middle School 23, no. 7 (May 2018): 360. http://dx.doi.org/10.5951/mathteacmiddscho.23.7.0360.

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YU, YUE, and DANIEL T. FINN. "Crescent Versus Rectangle." Dermatologic Surgery 36, no. 2 (February 2010): 171–76. http://dx.doi.org/10.1111/j.1524-4725.2009.01420.x.

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