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Athanasiadis, Christos A., and Francisco Santos. "On the topology of the Baues poset of polyhedral subdivisions." Topology 41, no. 3 (May 2002): 423–33. http://dx.doi.org/10.1016/s0040-9383(00)00044-6.
Bihan, Frédéric, and Ivan Soprunov. "Criteria for strict monotonicity of the mixed volume of convex polytopes." Advances in Geometry 19, no. 4 (October 25, 2019): 527–40. http://dx.doi.org/10.1515/advgeom-2018-0024.
Abstract Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ V(Q1, …, Qn). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1, …, Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 ∪ … ∪ Pn. In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.
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CHEUNG, YAM KI, and OVIDIU DAESCU. "FRÉCHET DISTANCE PROBLEMS IN WEIGHTED REGIONS." Discrete Mathematics, Algorithms and Applications 02, no. 02 (June 2010): 161–79. http://dx.doi.org/10.1142/s1793830910000644.
We discuss two versions of the Fréchet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases, we give algorithms for finding a (1 + ∊)-factor approximation of the Fréchet distance between two polygonal curves. We also consider the Fréchet distance between two polygonal curves among polyhedral obstacles in [Formula: see text] (1/∞ weighted region problem) and present a (1 + ∊)-factor approximation algorithm.
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Bishop, Joseph E., and N. Sukumar. "Polyhedral finite elements for nonlinear solid mechanics using tetrahedral subdivisions and dual-cell aggregation." Computer Aided Geometric Design 77 (February 2020): 101812. http://dx.doi.org/10.1016/j.cagd.2019.101812.
Makovicky, E., and T. Balić-Žunić. "New Measure of Distortion for Coordination Polyhedra." Acta Crystallographica Section B Structural Science 54, no. 6 (December 1, 1998): 766–73. http://dx.doi.org/10.1107/s0108768198003905.
A new global measure of distortion for coordination polyhedra is proposed, based on a comparison of the ratios Vs (circumscribed sphere)/Vp (polyhedron) calculated, respectively, for the real and ideal polyhedra of the same number of coordinated atoms which have the same circumscribed sphere. This formula can be simplified to υ (%) = 100[Vi (ideal) − Vr (real)]/Vi , where Vi and Vr are the volumes of the above-defined polyhedra. The global distortion can be combined with other polyhedral characteristics, e.g. with the eccentricity of the central atom in the polyhedron or with the degree of sphericity of the coordination sphere [Balić Zõunić & Makovicky (1996). Acta Cryst. B52, 78–81].Vs /Vp ratios are given for a number of ideal polyhedra, including several types of trigonal coordination prisms, with the aim of facilitating the distortion calculations. The application examples included in the paper are: complex sulfides based on PbS and SnS archetypes, coordination polyhedra of large cations in feldspars, a phase transformation in a monoclinic amphibole and the subdivision of structures isopointal to ilmenite.
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Locatelli, Marco. "Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes." Journal of Global Optimization 66, no. 4 (February 18, 2016): 629–68. http://dx.doi.org/10.1007/s10898-016-0418-4.
AbstractLet |K| be the metric polyhedron of a simplicial complex K. In this paper, we characterize a simplicial subdivision K′ of K preserving the metric topology for |K| as the one such that the set K′(0) of vertices of K′ is discrete in |K|. We also prove that two such subdivisions of K have such a common subdivision.
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Mitchell, Joseph S. B., David M. Mount, and Subhash Suri. "Query-Sensitive Ray Shooting." International Journal of Computational Geometry & Applications 07, no. 04 (August 1997): 317–47. http://dx.doi.org/10.1142/s021819599700020x.
Ray (segment) shooting is the problem of determining the first intersection between a ray (directed line segment) and a collection of polygonal or polyhedral obstacles. In order to process queries efficiently, the set of obstacle polyhedra is usually preprocessed into a data structure. In this paper we propose a query-sensitive data structure for ray shooting, which means that the performance of our data structure depends on the local geometry of obstacles near the query segment. We measure the complexity of the local geometry near the segment by a parameter called the simple cover complexity, denoted by scc(s) for a segment s. Our data structure consists of a subdivision that partitions the space into a collection of polyhedral cells, each of O(1) complexity. We answer a segment shooting query by wallking along the segment through the subdivision. Our first result is that, for any fixed dimension d, there exists a simple hierarchical subdivision in which no query segment s intersects more than O(scc(s)) cells. Our second result shows that in two dimensions such a subdivision of size O(n) can be constructed in time O(n log n), where n is the total number of vertices in all the obstacles.
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Qousini, Maysoon, Hasan Hdieb, and Eman Almuhur. "Applications of Locally Compact Spaces in Polyhedra: Dimension and Limits." WSEAS TRANSACTIONS ON MATHEMATICS 23 (February 27, 2024): 118–24. http://dx.doi.org/10.37394/23206.2024.23.14.
The study of applications of locally compact spaces in polyhedra in relation to their dimensions as well as homotopy and extension problems developed in the late 1940s and early 1950s under the leadership of mathematician. Many mathematicians studied application locally compact in polyhedra. A polyhedron can be obtained by subdivision, as a simplicial metric complex; thus, re-gluings of polyhedra can also be seen as simple complexes. Thus, the topology of a simplicial metric complex X is the topology quotient of the reattachment. The objective of this work is to shed light on the applications in polyhedra of locally compact spaces and to highlight the limits of these applications. A continuous application f of X in P defines a finite open overlay of X, and a partition of the unit subordinate to this overlay, f is homotopic to an application f ', obtained by composing the restriction to A, of an application of X in the KR polyhedron, and a simplistic application of a sub-polyhedron KR' in P. The problem of extension deserves to be elucidated to understand how it is possible to get around certain conceptual difficulties.
Dissertations / Theses on the topic "Polyhedral subdivisions":
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McDonald, Terry Lynn. "Piecewise polynomial functions on a planar region: boundary constraints and polyhedral subdivisions." Texas A&M University, 2003. http://hdl.handle.net/1969.1/3915.
Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines
of degree at most k forms a vector space Crk() Moreover, a nice way to study
Cr
k()is to embed n Rd+1, and form the cone b of with the origin. It turns
out that the set of splines on b is a graded module Cr b() over the polynomial ring
R[x1; : : : ; xd+1], and the dimension of Cr
k() is the dimension o
This dissertation follows the works of Billera and Rose, as well as Schenck and
Stillman, who each approached the study of splines from the viewpoint of homological
and commutative algebra. They both defined chain complexes of modules such that
Cr(b) appeared as the top homology module.
First, we analyze the effects of gluing planar simplicial complexes. Suppose
1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate
pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the
Mayer-Vietoris sequence to obtain a natural relationship between the spline modules
Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2).
Next, given a simplicial complex , we study splines which also vanish on the
boundary of. The set of all such splines is denoted by Cr(b). In this case, we will
discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b).
Finally, we consider splines which are defined on a polygonally subdivided region
of the plane. By adding only edges to to form a simplicial subdivision , we will
be able to find bounds for the dimensions of the vector spaces Cr
k() for k 0. In
particular, these bounds will be given in terms of the dimensions of the vector spaces
Cr
k() and geometrical data of both and .
This dissertation concludes with some thoughts on future research questions and
an appendix describing the Macaulay2 package SplineCode, which allows the study
of the Hilbert polynomials of the spline modules.
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Philippe, Eva. "Geometric realizations using regular subdivisions : construction of many polytopes, sweep polytopes, s-permutahedra." Electronic Thesis or Diss., Sorbonne université, 2024. http://www.theses.fr/2024SORUS079.
Cette thèse concerne trois problèmes de réalisations géométriques de structures combinatoires par des polytopes et des subdivisions polyédrales. Un polytope est l'enveloppe convexe d'un ensemble fini de points dans un espace euclidien R^d. Il est muni d'une structure combinatoire donnée par ses faces. Une subdivision est une collection de polytopes dont les faces s'intersectent correctement et dont l'union est convexe. Elle est régulière si elle peut être obtenue en prenant les faces inférieures d'un relèvement de ses sommets dans une dimension de plus.Nous présentons d'abord une nouvelle construction géométrique d'un grand nombre de polytopes combinatoirement distincts, de dimension et nombre de sommets fixés. Cette construction consiste à montrer que certains polytopes admettent un grand nombre de triangulations régulières. Elle nous permet d'améliorer la meilleure borne inférieure connue sur le nombre de types combinatoires de polytopes.Nous étudions ensuite les projections du permutoèdre, nommées polytopes de balayage (sweep polytopes) parce qu'elles modélisent les manières d'ordonner une configuration de points fixée en balayant l'espace par des hyperplans dans une direction constante. Nous introduisons également et étudions une abstraction combinatoire de ces structures : les matroïdes orientés de balayage, qui généralisent en dimension supérieure à 2 la théorie des suites admissibles de Goodman et Pollack.Enfin, nous proposons des réalisations géométriques de l'ordre s-faible, une structure combinatoire qui généralise l'ordre faible sur les permutations, paramétrée par un vecteur s à coordonnées entières strictement positives. En particulier, nous résolvons une conjecture de Ceballos et Pons en montrant que le s-permutoèdre peut être réalisé comme le graphe d'un complexe polytopal qui est une subdivision du permutoèdre This thesis concerns three problems of geometric realizations of combinatorial structures via polytopes and polyhedral subdivisions. A polytope is the convex hull of a finite set of points in a Euclidean space R^d. It is endowed with a combinatorial structure coming from its faces. A subdivision is a collection of polytopes whose faces intersect properly and such that their union is convex. It is regular if it can be obtained by taking the lower faces of a lifting of its vertices in one dimension higher.We first present a new geometric construction of many combinatorially different polytopes of fixed dimension and number of vertices. This construction relies on showing that certain polytopes admit many regular triangulations. It allows us to improve the best known lower bound on the number of combinatorial types of polytopes.We then study the projections of permutahedra, that we call sweep polytopes because they model the possible orderings of a fixed point configuration by hyperplanes that sweep the space in a constant direction. We also introduce and study a combinatorial abstraction of these structures: the sweep oriented matroids, that generalize Goodman and Pollack's theory of allowable sequences to dimensions higher than 2.Finally, we provide geometric realizations of the s-weak order, a combinatorial structure that generalizes the weak order on permutations, parameterized by a vector s with positive integer coordinates. In particular, we answer Ceballos and Pons conjecture that the s-weak order can be realized as the edge-graph of a polytopal complex that is moreover a subdivision of a permutahedron
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Moura, Phablo Fernando Soares. "Graph colorings and digraph subdivisions." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/45/45134/tde-23052017-100619/.
The vertex coloring problem is a classic problem in graph theory that asks for a partition of the vertex set into a minimum number of stable sets. This thesis presents our studies on three vertex (re)coloring problems on graphs and on a problem related to a long-standing conjecture on subdivision of digraphs. Firstly, we address the convex recoloring problem in which an arbitrarily colored graph G is given and one wishes to find a minimum weight recoloring such that each color class induces a connected subgraph of G. We show inapproximability results, introduce an integer linear programming (ILP) formulation that models the problem and present some computational experiments using a column generation approach. The k-fold coloring problem is a generalization of the classic vertex coloring problem and consists in covering the vertex set of a graph by a minimum number of stable sets in such a way that every vertex is covered by at least k (possibly identical) stable sets. We present an ILP formulation for this problem and show a detailed polyhedral study of the polytope associated with this formulation. The last coloring problem studied in this thesis is the proper orientation problem. It consists in orienting the edge set of a given graph so that adjacent vertices have different in-degrees and the maximum in-degree is minimized. Clearly, the in-degrees induce a partition of the vertex set into stable sets, that is, a coloring (in the conventional sense) of the vertices. Our contributions in this problem are on hardness and upper bounds for bipartite graphs. Finally, we study a problem related to a conjecture of Mader from the eighties on subdivision of digraphs. This conjecture states that, for every acyclic digraph H, there exists an integer f(H) such that every digraph with minimum out-degree at least f(H) contains a subdivision of H as a subdigraph. We show evidences for this conjecture by proving that it holds for some particular classes of acyclic digraphs. O problema de coloração de grafos é um problema clássico em teoria dos grafos cujo objetivo é particionar o conjunto de vértices em um número mínimo de conjuntos estáveis. Nesta tese apresentamos nossas contribuições sobre três problemas de coloração de grafos e um problema relacionado a uma antiga conjectura sobre subdivisão de digrafos. Primeiramente, abordamos o problema de recoloração convexa no qual é dado um grafo arbitrariamente colorido G e deseja-se encontrar uma recoloração de peso mínimo tal que cada classe de cor induza um subgrafo conexo de G. Mostramos resultados sobre inaproximabilidade, introduzimos uma formulação linear inteira que modela esse problema, e apresentamos alguns resultados computacionais usando uma abordagem de geração de colunas. O problema de k-upla coloração é uma generalização do problema clássico de coloração de vértices e consiste em cobrir o conjunto de vértices de um grafo com uma quantidade mínima de conjuntos estáveis de tal forma que cada vértice seja coberto por pelo menos k conjuntos estáveis (possivelmente idênticos). Apresentamos uma formulação linear inteira para esse problema e fazemos um estudo detalhado do politopo associado a essa formulação. O último problema de coloração estudado nesta tese é o problema de orientação própria. Ele consiste em orientar o conjunto de arestas de um dado grafo de tal forma que vértices adjacentes possuam graus de entrada distintos e o maior grau de entrada seja minimizado. Claramente, os graus de entrada induzem uma partição do conjunto de vértices em conjuntos estáveis, ou seja, induzem uma coloração (no sentido convencional) dos vértices. Nossas contribuições nesse problema são em complexidade computacional e limitantes superiores para grafos bipartidos. Finalmente, estudamos um problema relacionado a uma conjectura de Mader, dos anos oitenta, sobre subdivisão de digrafos. Esta conjectura afirma que, para cada digrafo acíclico H, existe um inteiro f(H) tal que todo digrafo com grau mínimo de saída pelo menos f(H) contém uma subdivisão de H como subdigrafo. Damos evidências para essa conjectura mostrando que ela é válida para classes particulares de digrafos acíclicos.
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Wang, Jiaxi. "PARAMETRIZATION AND SHAPE RECONSTRUCTION TECHNIQUES FOR DOO-SABIN SUBDIVISION SURFACES." UKnowledge, 2008. http://uknowledge.uky.edu/gradschool_theses/509.
This thesis presents a new technique for the reconstruction of a smooth surface from a set of 3D data points. The reconstructed surface is represented by an everywhere -continuous subdivision surface which interpolates all the given data points. And the topological structure of the reconstructed surface is exactly the same as that of the data points. The new technique consists of two major steps. First, use an efficient surface reconstruction method to produce a polyhedral approximation to the given data points. Second, construct a Doo-Sabin subdivision surface that smoothly passes through all the data points in the given data set. A new technique is presented for the second step in this thesis. The new technique iteratively modifies the vertices of the polyhedral approximation 1CM until a new control meshM, whose Doo-Sabin subdivision surface interpolatesM, is reached. It is proved that, for any mesh M with any size and any topology, the iterative process is always convergent with Doo-Sabin subdivision scheme. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully.
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Pham-Trong, Valérie. "Détermination géométrique de chemins géodésiques sur des surfaces de subdivision." Phd thesis, Université Joseph Fourier (Grenoble), 2001. http://www.theses.fr/2001GRE10112.
Un chemin géodésique entre deux points sur une surface de R³ est un plus court chemin local. Nous proposons deux méthodes de calcul de géodésiques qui ont l'originalité d'utiliser des outils de modélisation géométrique dans ce contexte de géométrie différentielle. La méthode de minimisation propose de travailler sur des surfaces paramétrées et d'étudier le problème en se plaçant dans l'espace des paramètres. Les courbes considérées y sont les courbes de Bézier et les courbes splines. Leurs points de contrôle constituent les variables par rapport auxquelles la longueur du chemin image sur la surface est minimisée. L'implémentation de cette méthode d'approximation et sa validation sont développées. La méthode de subdivision propose de travailler sur des surfaces de subdivision, limites d'une suite de réseaux générés par un schéma de subdivision. Une méthode itérative de calcul exact de chemin géodésique sur une surface polyédrique est développée. Celle-ci permet ainsi de calculer une suite de chemins géodésiques sur les surfaces polyédriques issues des réseaux de contrôle successifs. La convergence de cette suite de chemins géodésiques est traitée et de nombreux exemples sont présentés. Quelques applications sont enfin proposées : la génération de maillages surfaciques et la modélisation des fibres du myocarde pour l'imagerie médicale Geodesic paths between two points on a surface of R³ are local shortest paths. We propose two methods to compute them ; these ones are innovative because they use Computer Aided Geometric Design tools in this context of differential geometry. The minimisation method considers parametric surfaces and studies the problem in the parameter domain. Bezier and spline curves represent there the approximation class. Their control points are the variables for the minimization of the length of the image path on the surface. The implementation of this approximation method and its validation are developed. The subdivision method considers subdivision surfaces, limits of a sequence of control nets generated by a subdivision scheme. An iterative and exact method to compute geodesic paths on polyhedral surfaces is developed. This leads to the computation of a sequence of geodesic paths on the polyhedral surfaces associated to the successiv control nets. The convergence of the path sequence is discussed and we present results illustrated by examples. Some applications are finally given : surface mesh computation and myocardium fibres modelling in a medical context
Books on the topic "Polyhedral subdivisions":
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Popko, Edward. Divide spheres: Geodesics and the orderly subdivision of the sphere. Boca Raton: A K Peters/CRC Press, 2012.
Yamazaki, Shinji, Yoshihiro Yasumuro, and Masahiko Fuyuki. "Adaptive Polyhedral Subdivision for Image-based Lighting." In Service Robotics and Mechatronics, 177–82. London: Springer London, 2010. http://dx.doi.org/10.1007/978-1-84882-694-6_31.
Warren, Joe, and Henrik Weimer. "Averaging Schemes for Polyhedral Meshes." In Subdivision Methods for Geometric Design, 198–238. Elsevier, 2002. http://dx.doi.org/10.1016/b978-155860446-9/50009-0.
Jonckheere, Edmond A. "Piecewise-Linear Nyquist Map." In Algebraic and Differential Topology of Robust Stability, 113–23. Oxford University PressNew York, NY, 1997. http://dx.doi.org/10.1093/oso/9780195093018.003.0007.
Abstract Computing the crossover amounts to solving the Nyquist equation f(p) = 0 + jO. Assuming this equation is defined over the polyhedron D × Ω, the natural approach to implement an ad hoc solution in the realm of finite computation is to map the vertex set {ai} of the domain D × Ω to the complex plane, ai↦ bi= f (ai), and then work with the piecewise-linear extension of the vertex transformation ai↦bi. A typical feature of the resulting map is that the image of a simplex is a convex polygon, so that this piecewise-linear extension is not, in general, simplicial. However, an attractive feature of the piecewise-linear extension is that computation of the approximate crossover can be formulated as a linear program. Next, since the piecewise-linear extension lacks simplicial property, the standard fixup would be to compute a simplicial approximation to the piecewise-linear map. However, a conceptually more elegant approach is to make the piecewise-linear extension itself simplicial relative to refined subdivisions of both D × Ω and N. The reward is that crossover computation on a simplicial piecewise-linear map amounts to pure combinatorics, and the need for a linear program is obviated. It follows that, in a certain sense, simplicial approximation and linear programming are equivalent.
Conference papers on the topic "Polyhedral subdivisions":
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POSTNIKOV, ALEXANDER. "POSITIVE GRASSMANNIAN AND POLYHEDRAL SUBDIVISIONS." In International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0177.
Fortune, Steven. "Vertex-rounding a three-dimensional polyhedral subdivision." In the fourteenth annual symposium. New York, New York, USA: ACM Press, 1998. http://dx.doi.org/10.1145/276884.276897.
Mount, D. "Storing the subdivision of a polyhedral surface." In the second annual symposium. New York, New York, USA: ACM Press, 1986. http://dx.doi.org/10.1145/10515.10532.
Zachmann, Gabriel. "Real-Time and Exact Collision Detection for Interactive Virtual Prototyping." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/cie-4306.
Abstract Many companies have started to investigate Virtual Reality as a tool for evaluating digital mock-ups. One of the key functions needed for interactive evaluation is real-time collision detection. An algorithm for exact collision detection is presented which can handle arbitrary non-convex polyhedra efficiently. The approach attains its speed by a hierarchical adaptive space subdivision scheme, the BoxTree, and an associated divide-and-conquer traversal algorithm, which exploits the very special geometry of boxes. The traversal algorithm is generic, so it can be endowed with other semantics operating on polyhedra, e.g., distance computations. The algorithm is fairly simple to implement and it is described in great detail in an “ftp-able” appendix to facilitate easy implementation. Pre-computation of auxiliary data structures is very simple and fast. The efficiency of the approach is shown by timing results and two real-world digital mock-up scenarios.
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Dejdumrong, Natasha. "The determination of surface intersection using subdivision and Polyhedron Intersection Methods." In 2nd International Conference on Computer and Automation Engineering (ICCAE 2010). IEEE, 2010. http://dx.doi.org/10.1109/iccae.2010.5451590.
Liu, Ji-Bo, Zhi-Hong Wang, and Yue-Guan Yan. "A method of three-dimensional subdivision of arbitrary polyhedron by using pyramids." In 2016 5th International Conference on Measurement, Instrumentation and Automation (ICMIA 2016). Paris, France: Atlantis Press, 2016. http://dx.doi.org/10.2991/icmia-16.2016.73.
Skala, Vaclav. "Point-in-convex polygon and point-in-convex polyhedron algorithms with O(1) complexity using space subdivision." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2015 (ICNAAM 2015). Author(s), 2016. http://dx.doi.org/10.1063/1.4952270.
