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Journal articles on the topic 'Boolean of Polyhedral Solids'

Author: Grafiati

Published: 6 September 2023

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1

I.CHUBAREV, ALEXANDER. "ROBUST SET OPERATIONS ON POLYHEDRAL SOLIDS: A FIXED PRECISION APPROACH." International Journal of Computational Geometry & Applications 06, no. 02 (June 1996): 187–204. http://dx.doi.org/10.1142/s0218195996000137.

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An approach to the reliable boundary evaluation for polyhedral solids is proposed. The approach is based on the three ideas: an approximate evaluation of the Boolean operations is performed; precise calculations are performed at micro-level by using only exact numbers; triangulations are used for the boundary representation. Test examples illustrating efficiency of the approach are presented.

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2

JUAN-ARINYO, ROBERT, ÀLVAR VINACUA, and PERE BRUNET. "CLASSIFICATION OF A POINT WITH RESPECT TO A POLYHEDRON VERTEX." International Journal of Computational Geometry & Applications 06, no. 02 (June 1996): 157–67. http://dx.doi.org/10.1142/s0218195996000113.

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Algorithms for solid boolean operations are strongly based on classifying points with respect to solids. Several algorithms for solving the point-in- polyhedron problem in Brep schemes have been proposed in the literature. In this context, this paper describes an algorithm for the classification of an arbitrary point in the region dose to a polyhe-dron vertex. The algorithm is simple, has linear complexity and does not suffer from singularities. It performs better than previous algorithms, which would be too expensive in this particular case. The proposed algorithm is especially well suited for Brep schemes including spatial data structures for geometric data localization and searching, for operations on octree and extended octree structures, and for boundary evaluation of CSG trees with polyhedral primitives. Its use in the general point-in-polyhedron classification problem is also discussed. The algorithm is based on an extension to point in unbounded polygon of the kinetic framework of Guibas et al.

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3

Wang, C. C. L. "Approximate Boolean Operations on Large Polyhedral Solids with Partial Mesh Reconstruction." IEEE Transactions on Visualization and Computer Graphics 17, no. 6 (June 2011): 836–49. http://dx.doi.org/10.1109/tvcg.2010.106.

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4

Toriya, H., T. Takamura, T. Satoh, and H. Chiyokura. "Boolean operations for solids with free-form surfaces through polyhedral approximation." Visual Computer 7, no. 2-3 (March 1991): 87–96. http://dx.doi.org/10.1007/bf01901179.

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5

Menon, Jai, and Baining Guo. "Free-Form Modeling in Bilateral Brep and CSG Representation Schemes." International Journal of Computational Geometry & Applications 08, no. 05n06 (October 1998): 537–75. http://dx.doi.org/10.1142/s0218195998000278.

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This paper presents a unified approach for incorporating free-form solids in bilateral Brep and CSG representation schemes, by resorting to low-degree (quadratic, cubic) algebraic surface patches. We develop a general CSG solution that represents a free-form solid as a boolean combination of a direct term and a complicated delta term. This solution gives rise to the trunctet-subshell conditions, under which the delta term computation can be obviated. We use polyhedral smoothing to construct a Brep consisting of quadratic algebraic patches that meet with tangent-plane continuity, such that the trunctet-subshell conditions are guaranteed automatically. This guarantee is not currently available for cubic patches. The general CSG solution thus applies whenever trunctet-subshell conditions are violated, e.g. sometimes for cubic patches or sometimes for patches of any degree that are subject to shape control operations. Manifold solids of arbitrary topology can be represented in our dual representation system. Ensuing CSG constructs are parallel processed on the RayCasting Engine to support a wide range of solid modeling applications, including general sweeping, Minkowski operations, NC machining, and touch-sense probing.

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6

Landier, Sâm. "Boolean Operations on Arbitrary Polyhedral Meshes." Procedia Engineering 124 (2015): 200–212. http://dx.doi.org/10.1016/j.proeng.2015.10.133.

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7

Diazzi, Lorenzo, and Marco Attene. "Convex polyhedral meshing for robust solid modeling." ACM Transactions on Graphics 40, no. 6 (December 2021): 1–16. http://dx.doi.org/10.1145/3478513.3480564.

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We introduce a new technique to create a mesh of convex polyhedra representing the interior volume of a triangulated input surface. Our approach is particularly tolerant to defects in the input, which is allowed to self-intersect, to be non-manifold, disconnected, and to contain surface holes and gaps. We guarantee that the input surface is exactly represented as the union of polygonal facets of the output volume mesh. Thanks to our algorithm, traditionally difficult solid modeling operations such as mesh booleans and Minkowski sums become surprisingly robust and easy to implement, even if the input has defects. Our technique leverages on the recent concept of indirect geometric predicate to provide an unprecedented combination of guaranteed robustness and speed, thus enabling the practical implementation of robust though flexible solid modeling systems. We have extensively tested our method on all the 10000 models of the Thingi10k dataset, and concluded that no existing method provides comparable robustness, precision and performances.

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8

Verroust, A. "Visualization algorithm for CSG polyhedral solids." Computer-Aided Design 19, no. 10 (December 1987): 527–33. http://dx.doi.org/10.1016/0010-4485(87)90089-3.

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9

Hoffmann, C. M., J. E. Hopcroft, and M. J. Karasick. "Robust set operations on polyhedral solids." IEEE Computer Graphics and Applications 9, no. 6 (November 1989): 50–59. http://dx.doi.org/10.1109/38.41469.

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10

Landier, Sâm. "Boolean operations on arbitrary polygonal and polyhedral meshes." Computer-Aided Design 85 (April 2017): 138–53. http://dx.doi.org/10.1016/j.cad.2016.07.013.

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11

Flaquer, J., and J. L. Rodil. "Boolean operations based on the planar polyhedral representation." Computers & Graphics 12, no. 1 (January 1988): 59–64. http://dx.doi.org/10.1016/0097-8493(88)90008-8.

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12

Flaquer, J., and J. L. Rodil. "Boolean operations based on the planar polyhedral representation." Computer-Aided Design 20, no. 7 (September 1988): 424. http://dx.doi.org/10.1016/0010-4485(88)90241-2.

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13

Karasick, Michel. "The same-object problem for polyhedral solids." Computer Vision, Graphics, and Image Processing 46, no. 1 (April 1989): 22–36. http://dx.doi.org/10.1016/s0734-189x(89)80015-5.

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14

Karasick, Michael. "The same-object problem for polyhedral solids." Computer Vision, Graphics, and Image Processing 45, no. 2 (February 1989): 266–67. http://dx.doi.org/10.1016/0734-189x(89)90140-0.

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15

Burt, Justin L., Jose L. Elechiguerra, Jose Reyes-Gasga, J. Martin Montejano-Carrizales, and Miguel Jose-Yacaman. "Beyond Archimedean solids: Star polyhedral gold nanocrystals." Journal of Crystal Growth 285, no. 4 (December 2005): 681–91. http://dx.doi.org/10.1016/j.jcrysgro.2005.09.060.

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16

Zhao, Hanli, Charlie C. L. Wang, Yong Chen, and Xiaogang Jin. "Parallel and efficient Boolean on polygonal solids." Visual Computer 27, no. 6-8 (April 22, 2011): 507–17. http://dx.doi.org/10.1007/s00371-011-0571-1.

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17

Adams, Bart, and Philip Dutré. "Interactive boolean operations on surfel-bounded solids." ACM Transactions on Graphics 22, no. 3 (July 2003): 651–56. http://dx.doi.org/10.1145/882262.882320.

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18

Burdett, Jeremy K., and Gordon J. Miller. "Polyhedral clusters in solids. Electronic structure of pentlandite." Journal of the American Chemical Society 109, no. 13 (June 1987): 4081–91. http://dx.doi.org/10.1021/ja00247a039.

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19

Wills, J. M. "POLYHEDRAL MANIFOLDS WITH PROPERTIES OF THE PLATONIC SOLIDS." Annals of the New York Academy of Sciences 440, no. 1 Discrete Geom (May 1985): 224–29. http://dx.doi.org/10.1111/j.1749-6632.1985.tb14557.x.

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20

Kim, Yong Se, and D. J. Wilde. "A Convex Decomposition Using Convex Hulls and Local Cause of Its Non-Convergence." Journal of Mechanical Design 114, no. 3 (September 1, 1992): 459–67. http://dx.doi.org/10.1115/1.2926574.

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To exploit convexity, a non-convex object can be represented by a boolean combination of convex components. A convex decomposition method of polyhedral objects uses convex hulls and set difference operations. This decomposition, however, may not converge. In this article, we formalize this decomposition method and find local cause of non-convergence.

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21

Guo, Hao-Xiang, Yang Liu, Hao Pan, and Baining Guo. "Implicit Conversion of Manifold B-Rep Solids by Neural Halfspace Representation." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–15. http://dx.doi.org/10.1145/3550454.3555502.

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We present a novel implicit representation --- neural halfspace representation (NH-Rep), to convert manifold B-Rep solids to implicit representations. NH-Rep is a Boolean tree built on a set of implicit functions represented by the neural network, and the composite Boolean function is capable of representing solid geometry while preserving sharp features. We propose an efficient algorithm to extract the Boolean tree from a manifold B-Rep solid and devise a neural network-based optimization approach to compute the implicit functions. We demonstrate the high quality offered by our conversion algorithm on ten thousand manifold B-Rep CAD models that contain various curved patches including NURBS, and the superiority of our learning approach over other representative implicit conversion algorithms in terms of surface reconstruction, sharp feature preservation, signed distance field approximation, and robustness to various surface geometry, as well as a set of applications supported by NH-Rep.

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22

Strasser, Peter, Manuel Gliech, Stefanie Kuehl, and Tim Moeller. "Electrochemical processes on solid shaped nanoparticles with defined facets." Chemical Society Reviews 47, no. 3 (2018): 715–35. http://dx.doi.org/10.1039/c7cs00759k.

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23

KRISHNAN, S., D. MANOCHA, M. GOPI, T. CULVER, and J. KEYSER. "BOOLE: A BOUNDARY EVALUATION SYSTEM FOR BOOLEAN COMBINATIONS OF SCULPTURED SOLIDS." International Journal of Computational Geometry & Applications 11, no. 01 (February 2001): 105–44. http://dx.doi.org/10.1142/s0218195901000419.

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In this paper we describe a system, BOOLE, that generates the boundary representations (B-reps) of solids given as a CSG expression in the form of trimmed Bézier patches. The system makes use of techniques from computational geometry, numerical linear algebra and symbolic computation to generate the B-reps. Given two solids, the system first computes the intersection curve between the two solids using our surface intersection algorithm. Using the topological information of each solid, it computes various components within each solid generated by the intersection curve and their connectivity. The component classification step is performed by ray-shooting. Depending on the Boolean operation performed, appropriate components are put together to obtain the final solid. We also present techniques to parallelize this system on shared memory multiprocessor machines. The system has been successfully used to generate B-reps for a number of large industrial models including parts of a notional submarine storage and handling room (courtesy - Electric Boat Inc.) and Bradley fighting vehicle (courtesy - Army Research Labs). Each of these models is composed of over 8000 Boolean operations and is represented using over 50,000 trimmed Bézier patches. Our exact representation of the intersection curve and use of stable numerical algorithms facilitate an accurate boundary evaluation at every Boolean set operation and generation of topologically consistent solids.

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24

Krishnan, S., M. Gopi, D. Manocha, and M. Mine. "Interactive Boundary Computation of Boolean Combinations of Sculptured Solids." Computer Graphics Forum 16, no. 3 (September 1997): C67—C78. http://dx.doi.org/10.1111/1467-8659.00143.

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25

Krishnan, S., M. Gopi, D. Manocha, and M. Mine. "Interactive Boundary Computation of Boolean Combinations of Sculptured Solids." Computer Graphics Forum 16 (June 28, 2008): C67—C78. http://dx.doi.org/10.1111/1467-8659.16.3conferenceissue.8.

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26

Arbab, F. "Set models and Boolean operations for solids and assemblies." IEEE Computer Graphics and Applications 10, no. 6 (November 1990): 76–86. http://dx.doi.org/10.1109/38.62698.

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27

Pereira, André Maués Brabo, Marcos Chataignier de Arruda, Antônio Carlos de O. Miranda, William Wagner M. Lira, and Luiz Fernando Martha. "Boolean operations on multi-region solids for mesh generation." Engineering with Computers 28, no. 3 (June 28, 2011): 225–39. http://dx.doi.org/10.1007/s00366-011-0228-8.

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28

Xu, Jin Ting, Xiang Kui Zhang, and Shun Ke Wang. "Contour Offset Approach to Generate Tool Path for Polyhedral Surfaces Machining." Advanced Materials Research 314-316 (August 2011): 1638–41. http://dx.doi.org/10.4028/www.scientific.net/amr.314-316.1638.

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Polyhedral surfaces are used as representation model for CAM and process planning purposes because of its simplicity for data exchange and geometric computation. However, there is few tool path planning strategies for such surfaces but the iso-plane method. In this paper, contour parallel path are generated for three-axis ball-end milling. This tool path is based on a novel algorithm for offsetting curves on polyhedral surfaces presented in this paper. It reduces the task of removing complex interfering of offset curve from 3D surface to 2D plane by flattening mesh surface, and avoids costly 3D Boolean set operations and expensive distance calculation. This results in an efficient tool-path generation. Empirical examples illustrate the feasibility the proposed method.

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29

Smith, J. M., and N. A. Dodgson. "A topologically robust algorithm for Boolean operations on polyhedral shapes using approximate arithmetic." Computer-Aided Design 39, no. 2 (February 2007): 149–63. http://dx.doi.org/10.1016/j.cad.2006.11.003.

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30

Inui, Masatomo, Nobuyuki Umezu, and Yuuki Kitamura. "Visualizing sphere-contacting areas on automobile parts for ECE inspection." Journal of Computational Design and Engineering 2, no. 1 (December 6, 2014): 55–66. http://dx.doi.org/10.1016/j.jcde.2014.11.006.

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Abstract To satisfy safety regulations of Economic Commission for Europe (ECE), the surface regions of automobile parts must have a sufficient degree of roundness if there is any chance that they could contact a sphere of 50.0 mm radius (exterior parts) or 82.5 mm radius (interior parts). In this paper, a new offset-based method is developed to automatically detect the possible sphere-contacting shape of such parts. A polyhedral model that precisely approximates the part shape is given as input, and the offset shape of the model is obtained as the Boolean union of the expanded shapes of all surface triangles. We adopt a triple-dexel representation of the 3D model to enable stable and precise Boolean union computations. To accelerate the dexel operations in these Boolean computations, a new parallel processing method with a pseudo-list structure and axis-aligned bounding box is developed. The possible sphere-contacting shape of the part surface is then extracted from the offset shape as a set of points or a set of polygons.

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31

Sherbrooke, E. C., N. M. Patrikalakis, and E. Brisson. "An algorithm for the medial axis transform of 3D polyhedral solids." IEEE Transactions on Visualization and Computer Graphics 2, no. 1 (March 1996): 44–61. http://dx.doi.org/10.1109/2945.489386.

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32

Subramani, K. "On the Complexities of Selected Satisfiability and Equivalence Queries over Boolean Formulas and Inclusion Queries over Hulls." Journal of Applied Mathematics and Decision Sciences 2009 (July 20, 2009): 1–18. http://dx.doi.org/10.1155/2009/845804.

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This paper is concerned with the computational complexities of three types of queries, namely, satisfiability, equivalence, and hull inclusion. The first two queries are analyzed over the domain of CNF formulas, while hull inclusion queries are analyzed over continuous and discrete sets defined by rational polyhedra. Although CNF formulas can be represented by polyhedra over discrete sets, we analyze them separately on account of their distinct structure. In particular, we consider the NAESAT and XSAT versions of satisfiability over HornCNF, 2CNF, and Horn⊕2CNF formulas. These restricted families find applications in a number of practical domains. From the hull inclusion perspective, we are primarily concerned with the question of checking whether two succinct descriptions of a set of points are equivalent. In particular, we analyze the complexities of integer hull inclusion over 2SAT and Horn polyhedra. Hull inclusion problems are important from the perspective of deriving minimal descriptions of point sets. One of the surprising consequences of our work is the stark difference in complexities between equivalence problems in the clausal and polyhedral domains for the same polyhedral structure.

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33

Feito, F. R., C. J. Ogayar, R. J. Segura, and M. L. Rivero. "Fast and accurate evaluation of regularized Boolean operations on triangulated solids." Computer-Aided Design 45, no. 3 (March 2013): 705–16. http://dx.doi.org/10.1016/j.cad.2012.11.004.

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34

Ayala, D. "Boolean operations between solids and surfaces by octrees: models and algorithm." Computer-Aided Design 20, no. 8 (October 1988): 452–65. http://dx.doi.org/10.1016/0010-4485(88)90003-6.

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35

Ceulemans, A., L. F. Chibotaru, P. W. Fowler, and M. Szopa. "Symmetry extensions of Euler’s polyhedral theorem and the band theory of solids." Journal of Chemical Physics 110, no. 14 (April 8, 1999): 6916–26. http://dx.doi.org/10.1063/1.478597.

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36

SHAPIRO, VADIM. "WELL-FORMED SET REPRESENTATIONS OF SOLIDS." International Journal of Computational Geometry & Applications 09, no. 02 (April 1999): 125–50. http://dx.doi.org/10.1142/s0218195999000108.

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All point Membership Classification (PMC) algorithms on solids constructed using regularized set operations require representing and computing neighborhoods of a point with respect to the represented solid. Such computations involve some of the more sensitive and difficult algorithms in solid modeling. We establish the necessary and sufficient conditions under which set-theoretic constructions are well-formed so that PMC amounts to evaluating a ternary logic expression and does not require any neighborhood computations. We demonstrate well-formedness of any monotone set expression whose primitives form a simple arrangement in Ed, and of common representations for polyhedral objects constructed using 'decreasing convex hull' algorithms. Well-formed representations exist for every solid but not for every fixed collection of primitives. We also give a necessary and sufficient test for existence of such representations. Identifying and eliminating neighborhood computations whenever possible should lead to simpler, more efficient, and robust geometric algorithms that are also more suitable for hardware implementation. Finally, well-formed set representations can be trivially translated into representations of solids by real-valued implicit functions.

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37

Lee, Sang Han, Kyu-Yeul Lee, Yoonwhan Woo, and Kang-Soo Lee. "Feature-based multi-resolution modeling of solids using history-based Boolean operations — Part I: Theory of history-based boolean operations —." Journal of Mechanical Science and Technology 19, no. 2 (February 2005): 549–57. http://dx.doi.org/10.1007/bf02916177.

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38

Yan, Kedong, and Hong Seo Ryoo. "A multi-term, polyhedral relaxation of a 0–1 multilinear function for Boolean logical pattern generation." Journal of Global Optimization 74, no. 4 (June 25, 2018): 705–35. http://dx.doi.org/10.1007/s10898-018-0680-8.

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39

Ganter, M. A., and J. J. Uicker. "Dynamic Collision Detection Using Swept Solids." Journal of Mechanisms, Transmissions, and Automation in Design 108, no. 4 (December 1, 1986): 549–55. http://dx.doi.org/10.1115/1.3258768.

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The detection of collisions in an environment composed of two three-dimensional bodies traversing independent general three-dimensional trajectories is accomplished through the use of swept solids and solid modeling techniques. A swept solid represents the space volumetrically swept out by the motion of a given body along a given trajectory. A swept solid is created for each of the bodies in the given environment. Using the swept solids created for each body, calculations (solid modeling boolean intersections) can be performed to determine if these swept solids intersect. If the original bodies will collide while traversing their given trajectories, then their swept solids will statically interfere. Further, an object comprising the volume of the intersection can be created if the bodies do, in fact, interfere. This object can be thought of as the “volume of interference.” Enhancements to this technique provide for the formation of swept solids using relative motion. Through these enhancements, only one swept solid need be created since the absolute motions can be converted to motion of one body relative to another body. Therefore, intersection calculations may be performed between one relative swept solid and the original body.

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40

Tan, S. T., M. M. F. Yuen, and K. M. Yu. "Parameterized Object Design Using a Geometrical Construction Representation." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 207, no. 1 (February 1993): 21–30. http://dx.doi.org/10.1243/pime_proc_1993_207_058_02.

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This paper proposes a geometrical construction interpretation of variational geometry for achieving parameterized object definition. Vector geometry is used to represent the various kinds of geometrical entities while dimensions are defined to have directional sense. Geometrical constructions are then used to link up all geometrical entities and dimensions of a solid in a meaningful way, to represent the variational geometry properly. The geometrical construction scheme is implemented in a prototype modeller using polyhedral solids as examples.

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41

Feito, Francisco R., and Juan C. Torres. "Boundary representation of polyhedral heterogeneous solids in the context of a graphic object algebra." Visual Computer 13, no. 2 (March 27, 1997): 64–77. http://dx.doi.org/10.1007/s003710050090.

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42

NAKAMURA, Hiroyuki, Masatake HIGASHI, and Mamoru HOSAKA. "Robust Interference Calculation of Polyhedral Solids by Symbolic Calculation Using Face Names (1st Report)." Journal of the Japan Society for Precision Engineering 63, no. 4 (1997): 515–19. http://dx.doi.org/10.2493/jjspe.63.515.

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43

NAKAMURA, Hiroyuki, Masatake HIGASHI, and Mamoru HOSAKA. "Robust Interference Calculation of Polyhedral Solids by Symbolic Calculation Using Face Names (2nd Report)." Journal of the Japan Society for Precision Engineering 64, no. 1 (1998): 106–10. http://dx.doi.org/10.2493/jjspe.64.106.

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44

Scharfe, Sandra, and Thomas F. Fässler. "Polyhedral nine-atom clusters of tetrel elements and intermetalloid derivatives." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1915 (March 28, 2010): 1265–84. http://dx.doi.org/10.1098/rsta.2009.0270.

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Homoatomic polyanions have the basic capability for a bottom-up synthesis of nanostructured materials. Therefore, the chemistry and the structures of polyhedral nine-atom clusters of tetrel elements [E 9 ] 4− is highlighted. The nine-atom Zintl ions are available in good quantities for E = Si–Pb as binary alkali metal (A) phases of the composition A 4 E 9 or A 12 E 17 . Dissolution or extraction of the neat solids with aprotic solvents and crystallization with alkali metal-sequestering molecules or crown ethers leads to a large variety of structures containing homoatomic clusters with up to 45 E atoms. Cluster growth occurs via oxidative coupling reactions. The clusters can further act as a donor ligand in transition metal complexes, which is a first step to the formation of bimetallic clusters. The structures and some nuclear magnetic resonance spectroscopic properties of these so-called intermetalloid clusters are reviewed, with special emphasis on tetrel clusters that are endohedrally filled with transition metal atoms.

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45

APT, KRZYSZTOF R., and ERIC MONFROY. "Constraint programming viewed as rule-based programming." Theory and Practice of Logic Programming 1, no. 6 (November 2001): 713–50. http://dx.doi.org/10.1017/s1471068401000072.

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We study here a natural situation when constraint programming can be entirely reduced to rule-based programming. To this end we explain first how one can compute on constraint satisfaction problems using rules represented by simple first-order formulas. Then we consider constraint satisfaction problems that are based on predefined, explicitly given constraints. To solve them we first derive rules from these explicitly given constraints and limit the computation process to a repeated application of these rules, combined with labeling. We consider two types of rule here. The first type, that we call equality rules, leads to a new notion of local consistency, called rule consistency that turns out to be weaker than arc consistency for constraints of arbitrary arity (called hyper-arc consistency in Marriott & Stuckey (1998)). For Boolean constraints rule consistency coincides with the closure under the well-known propagation rules for Boolean constraints. The second type of rules, that we call membership rules, yields a rule-based characterization of arc consistency. To show feasibility of this rule-based approach to constraint programming, we show how both types of rules can be automatically generated, as CHR rules of Frühwirth (1995). This yields an implementation of this approach to programming by means of constraint logic programming. We illustrate the usefulness of this approach to constraint programming by discussing various examples, including Boolean constraints, two typical examples of many valued logics, constraints dealing with Waltz's language for describing polyhedral scenes, and Allen's qualitative approach to temporal logic.

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46

Gosselin, C. M., and D. Gagnon-Lachance. "Expandable Polyhedral Mechanisms Based on Polygonal One-Degree-of-Freedom Faces." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220, no. 7 (July 1, 2006): 1011–18. http://dx.doi.org/10.1243/09544062jmes174.

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In this article, a new family of expandable mechanisms is presented. The proposed mechanisms are expandable polyhedra built using one-degree-of-freedom (one-DOF) planar linkages. The latter planar linkages have the shape of polygons and can be expanded while preserving their shape in any of their configurations. The planar mechanisms are used to form the faces of a polyhedron. They are assembled using spherical joints at the vertices of the polyhedron. The result is a one-DOF movable polyhedron which can be expanded while preserving its shape. The application of the principle on regular polyhedra is first presented. For the five Platonic solids, theoretical maximum expansion ratios are computed, simulation results are given, and two prototypes are shown. Then, two additional examples are provided to illustrate the application of the principle to irregular polyhedra.

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47

Atserias, Albert, Anuj Dawar, and Joanna Ochremiak. "On the Power of Symmetric Linear Programs." Journal of the ACM 68, no. 4 (July 28, 2021): 1–35. http://dx.doi.org/10.1145/3456297.

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We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs, while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991), showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.

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48

Glicksman, Martin E. "Capillarity-Mediated Grain Growth in 3-D." Materials Science Forum 467-470 (October 2004): 1025–32. http://dx.doi.org/10.4028/www.scientific.net/msf.467-470.1025.

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Space-filling in kinetically active 3-d network structures, such as polycrystalline solids at high temperatures, is treated using topological methods. The theory developed represents individual network elements—the polyhedral cells or grains—as a set of objects called average N-hedra, where N, the topological class, equals the number of contacting neighbors in the network. Average N-hedra satisfy network topological averages for the dihedral angles and quadrajunction vertex angles, and, most importantly, act as “proxies” for real irregular polyhedral grains with equivalent topology. The analysis provided in this paper describes the energetics and kinetics of grains represented as average N-hedra as a function of their topological class. The new approach provides a quantitative basis for constructing more accurate models of three-dimensional grain growth. As shown, the availability of rigorous mathematical relations for the curvatures, areas, volumes, free energies, and rates of volume change provides precise predictions to test simulations of the behavior 3-d networks, and to guide quantitative experiments on microstructure evolution in three-dimensional polycrystals.

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García, Á. L., J. Ruiz de Miras, and F. R. Feito. "Evaluation of Boolean operations between free-form solids using extended simplicial chains and PN triangles." Visual Computer 27, no. 6-8 (April 15, 2011): 531–41. http://dx.doi.org/10.1007/s00371-011-0566-y.

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Dechant, Pierre-Philippe. "Platonic solids generate their four-dimensional analogues." Acta Crystallographica Section A Foundations of Crystallography 69, no. 6 (September 12, 2013): 592–602. http://dx.doi.org/10.1107/s0108767313021442.

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This paper shows how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone.Viathe Cartan–Dieudonné theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, theF4root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_{1}\oplus I_{2}(n) which induces I_{2}(n)\oplus I_{2}(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

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