Academic literature on the topic 'Rectangle'

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Journal articles on the topic "Rectangle"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Rectangle"

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Mahmood, Abdullah-Al. "Approximation Algorithms for Rectangle Piercing Problems." Thesis, University of Waterloo, 2005. http://hdl.handle.net/10012/1025.

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Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our effort to designing approximation algorithms for unit-height rectangles. Our e-approximation scheme requires nO(1/ε²) time. We also consider the problem with restrictions like bounding the depth of a point and the width of the rectangles. The approximation schemes for these two cases take nO(1/ε) time. We also show how to maintain a factor 2 approximation of the piercing set in O(log n) amortized time in an insertion-only scenario.
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Song, In-Ok. "Infrared emission bands of the Red Rectangle." Thesis, University of Nottingham, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.416304.

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Chung, Yau-lin, and 鍾有蓮. "Optimality and approximability of the rectangle covering problem." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B30294873.

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Forsman, Anna. "-those complete strangers- an investigation of the rectangle." Thesis, Högskolan i Borås, Institutionen Textilhögskolan, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:hb:diva-20702.

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An report about investigating the rectangular shape in the relation between the stiff and the soft in drapings. The investigation have been made in the field fashion and garments.
Program: Modedesignutbildningen
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Thomas, Joshua David. "Spectroscopic Analysis and Modeling of the Red Rectangle." University of Toledo / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1341345222.

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LeBlanc, Denyse I. "Congelation d'un aliment ayant la forme d'un parallelepipede rectangle." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=63893.

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Silvanus, Jannik [Verfasser]. "Improved Cardinality Bounds for Rectangle Packing Representations / Jannik Silvanus." Bonn : Universitäts- und Landesbibliothek Bonn, 2019. http://d-nb.info/1188726226/34.

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Theise, Helena. "F ME F YOU : an investigation of the expressional potential of rectangular pattern construction in relation to print." Thesis, Högskolan i Borås, Akademin för textil, teknik och ekonomi, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:hb:diva-11118.

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This work is exploring the rectangle as a pattern construction. It is the most recognised geometric shape, can it still provide us with new expressions in fashion? This project is conducted through clear restrictions in the method, and through draping translated into garments through flat pattern construction. The result is a collection with a complex expression, mixing poetic shapes with playful prints full of contrast, which signifes harmony but does not follow the classical notions of beauty. The value of this work lies in the finding of new expressions in fashion, proposing that it is of utmost importance to challenge what we think we know to be true.
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Stylianopoulos, Nikalaos Stavros. "A domain decomposition method for numerical conformal mapping onto a rectangle." Thesis, Brunel University, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.257545.

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Topalović, Radmila. "Infrared and optical emission bands of the Red Rectangle and other objects." Thesis, University of Nottingham, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.438417.

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Books on the topic "Rectangle"

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Jean, Robertson J., ed. ?Un cuadrado? ! Un rectangulo! =: A square? A rectangle! Vero Beach, FL: Rourke Pub., 2009.

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Rectangle of light. Aylmer, Quebec: Proof Press, 1996.

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Plip, la planète rectangle. [Paris]: Delcourt, 1995.

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A square? A rectangle! Vero Beach, FL: Rourke Pub., 2009.

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Inside the magic rectangle. London: V. Gollancz, 1995.

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Hugo et le rectangle. [Paris]: Hachette jeunesse, 1993.

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Jean, Robertson J., ed. A square? a rectangle! Vero Beach, FL: Rourke Pub., 2009.

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Oberto, Varinia. Un rectangle de plaisir: Roman. Paris: Presses de la Renaissance, 1988.

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Atallah, Mikhail J. Output-sensitive hidden surface elimination for rectangles. [Moffett Field, Calif.?]: Research Institute for Advanced Computer Science, 1989.

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Atallah, Mikhail J. Output-sensitive hidden surface elimination for rectangles. [Moffett Field, Calif.?]: Research Institute for Advanced Computer Science, 1989.

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Book chapters on the topic "Rectangle"

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Zambon, Giulio. "Implementing “Rectangle”." In Sudoku Programming with C, 133–48. Berkeley, CA: Apress, 2015. http://dx.doi.org/10.1007/978-1-4842-0995-0_12.

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Golub, Spencer. "Page (Rectangle)." In Heidegger and Future Presencing (The Black Pages), 1–41. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-31889-5_1.

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Kwok, Sun. "Red Rectangle." In Encyclopedia of Astrobiology, 2158–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-44185-5_5076.

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Stemkoski, Lee, and Evan Leider. "Rectangle Destroyer." In Game Development with Construct 2, 103–14. Berkeley, CA: Apress, 2017. http://dx.doi.org/10.1007/978-1-4842-2784-8_8.

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Shekhar, Shashi, and Hui Xiong. "Rectangle, Hyper-." In Encyclopedia of GIS, 955. Boston, MA: Springer US, 2008. http://dx.doi.org/10.1007/978-0-387-35973-1_1093.

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Loryś, Krzysztof, and Katarzyna Paluch. "Rectangle Tiling." In Approximation Algorithms for Combinatorial Optimization, 206–13. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/3-540-44436-x_21.

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Chazal, Frédéric, Vin de Silva, Marc Glisse, and Steve Oudot. "Rectangle Measures." In The Structure and Stability of Persistence Modules, 31–66. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-42545-0_3.

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Kwok, Sun. "Red Rectangle." In Encyclopedia of Astrobiology, 1–3. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-27833-4_5076-6.

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Wood, Jordan. "Minimum Bounding Rectangle." In Encyclopedia of GIS, 660–61. Boston, MA: Springer US, 2008. http://dx.doi.org/10.1007/978-0-387-35973-1_783.

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Yeung, Daniel S., Ian Cloete, Daming Shi, and Wing W. Y. Ng. "Hyper-Rectangle Model." In Natural Computing Series, 25–27. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-02532-7_3.

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Conference papers on the topic "Rectangle"

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Kovács, Kristóf, and Boglárka G.-Tóth. "Rectangle covering." In PROCEEDINGS LEGO – 14TH INTERNATIONAL GLOBAL OPTIMIZATION WORKSHOP. Author(s), 2019. http://dx.doi.org/10.1063/1.5090003.

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Zhu, Shenglong, Scott J. Emrich, and Danny Z. Chen. "Predicting Local Inversions Using Rectangle Clustering and Representative Rectangle Prediction." In 2018 IEEE International Conference on Bioinformatics and Biomedicine (BIBM). IEEE, 2018. http://dx.doi.org/10.1109/bibm.2018.8621190.

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Dobkin, David P., and Dimitrios Gunopulos. "Computing the rectangle discrepancy." In the tenth annual symposium. New York, New York, USA: ACM Press, 1994. http://dx.doi.org/10.1145/177424.178098.

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Lin, Sching L., and Jonathan Allen. "Minplex---a compactor that minimizes the bounding rectangle and individual rectangles in a layout." In the 23rd ACM/IEEE conference. New York, New York, USA: ACM Press, 1986. http://dx.doi.org/10.1145/318013.318033.

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Lin, S. L., and J. Allen. "Minplex - A Compactor that Minimizes the Bounding Rectangle and Individual Rectangles in a Layout." In 23rd ACM/IEEE Design Automation Conference. IEEE, 1986. http://dx.doi.org/10.1109/dac.1986.1586078.

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Jingyu Yang and Zhongyu Jiang. "Rectangle fitting via quadratic programming." In 2015 IEEE 17th International Workshop on Multimedia Signal Processing (MMSP). IEEE, 2015. http://dx.doi.org/10.1109/mmsp.2015.7340875.

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Ralevic, Nebojsa M., Slobodan Drazic, and Radovan Obradovic. "The Hough transformation of rectangle." In 2008 6th International Symposium on Intelligent Systems and Informatics (SISY 2008). IEEE, 2008. http://dx.doi.org/10.1109/sisy.2008.4664906.

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Bazin, Jean-Charles, Inso Kweon, Cedric Demonceaux, and Pascal Vasseur. "Rectangle Extraction in Catadioptric Images." In 2007 IEEE 11th International Conference on Computer Vision. IEEE, 2007. http://dx.doi.org/10.1109/iccv.2007.4409208.

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Wang, Cheng. "Fast Method for Rectangle Detection." In 2016 6th International Conference on Machinery, Materials, Environment, Biotechnology and Computer. Paris, France: Atlantis Press, 2016. http://dx.doi.org/10.2991/mmebc-16.2016.180.

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Yim, H., and A. C. Butler. "Checking Geometric Constraints in Layout Design Using Primitives." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/cie-1657.

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Abstract The use of geometric primitives provides representation of objects in layout design at an appropriate level of abstraction, and it allows the use of two new algorithms which permit improvements in computational speed for constraint checking. These algorithms detect intersections between two rectangles and between a rectangle and circle with improved computational efficiency. The use of this pair of algorithms is demonstrated on test problems executed on a parallel computer, and conclusions are drawn regarding the use of geometric primitives for constraint checking in layout design.
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Reports on the topic "Rectangle"

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Silver, G. L. Operational equations for the five-point rectangle. Office of Scientific and Technical Information (OSTI), September 1993. http://dx.doi.org/10.2172/10185763.

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Walker, J. S. Liquid-metal MHD flow in a duct whose cross section changes from a rectangle to a trapezoid, with applications in fusion blanket designs. Office of Scientific and Technical Information (OSTI), April 1986. http://dx.doi.org/10.2172/5409989.

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Danielson, Thomas. A MONTE CARLO RECTANGLE PACKING ALGORITHM FOR IDENTIFYING LIKELY SPATIAL DISTRIBUTIONS OF FINAL CLOSURE CAP SUBSIDENCE IN THE E-AREA LOW-LEVEL WASTE FACILITY. Office of Scientific and Technical Information (OSTI), October 2019. http://dx.doi.org/10.2172/1571419.

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Yager, Robert J. Creating, Positioning, and Rotating Rectangles Using C++. Fort Belvoir, VA: Defense Technical Information Center, August 2013. http://dx.doi.org/10.21236/ada591373.

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Shiogi, Ann. Connected Painted Rectangles Experiments in Quantitative Shape and Contrasting Elements. Portland State University Library, January 2000. http://dx.doi.org/10.15760/etd.6553.

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Hoover, Donald R. Different Algorithms for Obtaining Upper Bounds to Multivariate Normal Areas Outside of Origin Centered Rectangles Using Joint Marginal Probabilities. Fort Belvoir, VA: Defense Technical Information Center, September 1988. http://dx.doi.org/10.21236/ada199772.

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