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Journal articles on the topic 'Polyhedral subdivisions'

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Published: 7 July 2024

Last updated: 7 July 2024

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1

Tawarmalani, Mohit, Jean-Philippe P. Richard, and Chuanhui Xiong. "Explicit convex and concave envelopes through polyhedral subdivisions." Mathematical Programming 138, no. 1-2 (July 31, 2012): 531–77. http://dx.doi.org/10.1007/s10107-012-0581-4.

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2

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.

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3

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.

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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.

4

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.

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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.

5

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.

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6

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.

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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.

7

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.

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8

Mine, Kotaro, and Katsuro Sakai. "Subdivisions of Simplicial Complexes Preserving the Metric Topology." Canadian Mathematical Bulletin 55, no. 1 (March 1, 2012): 157–63. http://dx.doi.org/10.4153/cmb-2011-055-7.

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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.

9

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.

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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.

10

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.

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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.

11

Dutour Sikirić, Mathieu, Alexey Garber, Achill Schürmann, and Clara Waldmann. "The complete classification of five-dimensional Dirichlet–Voronoi polyhedra of translational lattices." Acta Crystallographica Section A Foundations and Advances 72, no. 6 (October 3, 2016): 673–83. http://dx.doi.org/10.1107/s2053273316011682.

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This paper reports on the full classification of Dirichlet–Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. A complete list is obtained of 110 244 affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet–Voronoi polyhedra. Using a refinement of corresponding secondary cones, 181 394 contraction types are obtained. The paper gives details of the computer-assisted enumeration, which was verified by three independent implementations and a topological mass formula check.

12

Nasri, Ahmad H. "Polyhedral subdivision methods for free-form surfaces." ACM Transactions on Graphics 6, no. 1 (January 1987): 29–73. http://dx.doi.org/10.1145/27625.27628.

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13

Mount, David M. "Storing the subdivision of a polyhedral surface." Discrete & Computational Geometry 2, no. 2 (June 1987): 153–74. http://dx.doi.org/10.1007/bf02187877.

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14

Fortune, S. "Vertex-Rounding a Three-Dimensional Polyhedral Subdivision." Discrete & Computational Geometry 22, no. 4 (December 1999): 593–618. http://dx.doi.org/10.1007/pl00009480.

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15

Ortiz, Carlos, Adriana Lara, Jesús González, and Ayse Borat. "A Randomized Greedy Algorithm for Piecewise Linear Motion Planning." Mathematics 9, no. 19 (September 23, 2021): 2358. http://dx.doi.org/10.3390/math9192358.

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We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

16

Rushton, Brian. "Constructing subdivision rules from polyhedra with identifications." Algebraic & Geometric Topology 12, no. 4 (October 27, 2012): 1961–92. http://dx.doi.org/10.2140/agt.2012.12.1961.

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17

Peters, Jörg, and Ulrich Reif. "The simplest subdivision scheme for smoothing polyhedra." ACM Transactions on Graphics 16, no. 4 (October 1997): 420–31. http://dx.doi.org/10.1145/263834.263851.

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18

Max, Nelson. "Consistent Subdivision of Convex Polyhedra into Tetrahedra." Journal of Graphics Tools 6, no. 3 (January 2001): 29–36. http://dx.doi.org/10.1080/10867651.2001.10487543.

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19

Ginkel, I., J. Peters, and G. Umlauf. "Normals of subdivision surfaces and their control polyhedra." Computer Aided Geometric Design 24, no. 2 (February 2007): 112–16. http://dx.doi.org/10.1016/j.cagd.2006.10.005.

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20

CHEN, JINDONG, and YIJIE HAN. "SHORTEST PATHS ON A POLYHEDRON, Part I: COMPUTING SHORTEST PATHS." International Journal of Computational Geometry & Applications 06, no. 02 (June 1996): 127–44. http://dx.doi.org/10.1142/s0218195996000095.

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We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for processing queries of shortest paths between the source point and any destination point. Our algorithm uses a new approach which deviates from the conventional “continuous Dijkstra” technique. Our algorithm has time complexity O(n2) and space complexity Θ(n).

21

Boscardín, Liliana Beatriz, Liliana Raquel Castro, and Silvia Mabel Castro. "Haar-LikeWavelets over Tetrahedra." Journal of Computer Science and Technology 17, no. 02 (October 1, 2017): e13. http://dx.doi.org/10.24215/16666038.17.e13.

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In this paper we define a Haar-like wavelets basis that form a basis for L2(T,S,μ), μ being the Lebesgue measure and S the σ -algebra of all tetrahedra generated from a subdivision method of the T tetrahedron. As 3D objects are, in general, modeled by tetrahedral grids, this basis allows the multiresolution representation of scalar functions defined on polyhedral volumes, like colour, brightness, density and other properties of an 3D object.

22

Gardner, R. J., and M. Kallay. "Subdivision algorithms and the kernel of a polyhedron." Discrete & Computational Geometry 8, no. 4 (December 1992): 417–27. http://dx.doi.org/10.1007/bf02293056.

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23

Zheng, H.-C., G.-H. Peng, Z.-L. Ye, and L.-L. Pan. "A new ternary interpolatory subdivision scheme for polyhedral meshes with arbitrary topology." Journal of Physics: Conference Series 96 (February 1, 2008): 012072. http://dx.doi.org/10.1088/1742-6596/96/1/012072.

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24

Mehlhorn, Kurt, and Michael Seel. "Infimaximal Frames: A Technique for Making Lines Look Like Segments." International Journal of Computational Geometry & Applications 13, no. 03 (June 2003): 241–55. http://dx.doi.org/10.1142/s0218195903001141.

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Many geometric algorithms that are usually formulated for points and segments generalize easily to inputs also containing rays and lines. The sweep algorithm for segment intersection is a prototypical example. Implementations of such algorithms do, in general, not extend easily. For example, segment endpoints cause events in sweep line algorithms, but lines have no endpoints. We describe a general technique, which we call infimaximal frames, for extending implementations to inputs also containing rays and lines. The technique can also be used to extend implementations of planar subdivisions to subdivisions with many unbounded faces. We have used the technique successfully in generalizing a sweep algorithm designed for segments to rays and lines and also in an implementation of planar Nef polyhedra.14,1 Our implementation is based on concepts of generic programming in C++ and the geometric data types provided by the C++ Computational Geometry Algorithms Library (CGAL).

25

Patrikalakis, N. M., and P. V. Prakash. "Surface Intersections for Geometric Modeling." Journal of Mechanical Design 112, no. 1 (March 1, 1990): 100–107. http://dx.doi.org/10.1115/1.2912565.

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Evaluation of planar algebraic curves arises in the context of intersections of algebraic surfaces with piecewise continuous rational polynomial parametric surface patches useful in geometric modeling. We address a method of evaluating these curves of intersection that combines the advantageous features of analytic representation of the governing equation of the algebraic curve in the Bernstein basis within a rectangular domain, adaptive subdivision and polyhedral faceting techniques, and the computation of turning and singular points, to provide the basis for a reliable and efficient solution procedure. Using turning and singular points, the intersection problem can be partitioned into subdomains that can be processed independently and which involve intersection segments that can be traced with faceting methods. This partitioning and the tracing of individual segments is carried out using an adaptive subdivision algorithm for Bezier/B-spline surfaces followed by Newton correction of the approximation. The method has been successfully tested in tracing complex algebraic curves and in solving actual intersection problems with diverse features.

26

Wenninger, Magnus J., and Peter W. Messer. "Patterns On The Spherical Surface." International Journal of Space Structures 11, no. 1-2 (April 1996): 183–92. http://dx.doi.org/10.1177/026635119601-224.

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This article is a sequel to that which the first author presented in Vol. 5, Nos. 3 & 4 (1990), of this Journal. In that article he displayed his work on patterns derived from skew networks. Here he extends this to patterns derived from subdivisions of the faces of other regular and semiregular polyhedra as these are projected onto the spherical surface. Some mathematical formulas are presented by the second author which can readily be used in a programmable calculator to obtain arcs, chord factors and radian measures for any frequency of subdivision and any suitable spherical radius. The first author made the papercraft models shown in the photos. The article ends with words of encouragement for artists, architects and engineers to use patterns in ornamental designs and in architectural projects. Since the appearance of the article1 in this Journal the first author has been extensively engaged in developing new ways for making spherical models. He has also taken up an interest in using geometric patterns in a variety of designs. In his efforts to achieve some artistically beautiful effects, he has developed better ways to construct models still using only paper card stock for material as described in his published works2,3. A good successful model, however, can only be made when all the mathematical calculations have preceded the construction. Hence careful planning and preparatory drawings must be done before actual construction begins. The geodesic mathematics has all been done and is available for applications by anyone who wishes to study it. See the reference sources at the end of this article. The mathematics can often be very involved and hence often difficult to use.

27

Peters, J., and X. Wu. "The distance of a subdivision surface to its control polyhedron." Journal of Approximation Theory 161, no. 2 (December 2009): 491–507. http://dx.doi.org/10.1016/j.jat.2008.10.012.

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28

Meng, Li, Xiaochong Tong, Shuaibo Fan, Chengqi Cheng, Bo Chen, Weiming Yang, and Kaihua Hou. "A Universal Generating Algorithm of the Polyhedral Discrete Grid Based on Unit Duplication." ISPRS International Journal of Geo-Information 8, no. 3 (March 19, 2019): 146. http://dx.doi.org/10.3390/ijgi8030146.

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Based on the analysis of the problems in the generation algorithm of discrete grid systems domestically and abroad, a new universal algorithm for the unit duplication of a polyhedral discrete grid is proposed, and its core is “simple unit replication + effective region restriction”. First, the grid coordinate system and the corresponding spatial rectangular coordinate system are established to determine the rectangular coordinates of any grid cell node. Then, the type of the subdivision grid system to be calculated is determined to identify the three key factors affecting the grid types, which are the position of the starting point, the length of the starting edge, and the direction of the starting edge. On this basis, the effective boundary of a multiscale grid can be determined and the grid coordinates of a multiscale grid can be obtained. A one-to-one correspondence between the multiscale grids and subdivision types can be established. Through the appropriate rotation, translation and scaling of the multiscale grid, the node coordinates of a single triangular grid system are calculated, and the relationships between the nodes of different levels are established. Finally, this paper takes a hexagonal grid as an example to carry out the experiment verifications by converting a single triangular grid system (plane) directly to a single triangular grid with a positive icosahedral surface to generate a positive icosahedral surface grid. The experimental results show that the algorithm has good universality and can generate the multiscale grid of an arbitrary grid configuration by adjusting the corresponding starting transformation parameters.

29

Etzion, Michal, and Ari Rappoport. "Computing Voronoi skeletons of a 3-D polyhedron by space subdivision." Computational Geometry 21, no. 3 (March 2002): 87–120. http://dx.doi.org/10.1016/s0925-7721(01)00056-6.

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30

Abdul Karim, Samsul Ariffin, Faheem Khan, Ghulam Mustafa, Aamir Shahzad, and Muhammad Asghar. "An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes." Mathematics 11, no. 11 (May 25, 2023): 2449. http://dx.doi.org/10.3390/math11112449.

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Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.

31

Geng, Qing Jia, Xi Juan Guo, and Ya Zhang. "Research on Exact Minkowski Sum Algorithm of Convex Polyhedron Based on Direct Mapping." Advanced Materials Research 225-226 (April 2011): 377–80. http://dx.doi.org/10.4028/www.scientific.net/amr.225-226.377.

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Minkowski sum has become an effective method in collision detection problem, which is a branch of computation geometry. Separated from the previous algorithm based on the traditional Gaussian Map, a new algorithm of computing exact Minkowski sum of convex polyhedron is proposed based on direct mapping method in the paper, and the correctness of direct mapping method is testified. The algorithm mapping the convex polyhedron into the bottom of regular tetrahedron according to the definition of Regular Tetrahedron Mapping and Point Projection, so the problem become form 3D to 2D. Comparing with the previous algorithm, the algorithm posed in the paper establishes mapping from 3D to 2D directivity, and only compute the overlay of one pair of planar subdivision. So, the algorithm’s executing efficiency has been improved in compare with the previous algorithm.

32

Augsdörfer, U. H., N. A. Dodgson, and M. A. Sabin. "Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra." Computer Aided Geometric Design 28, no. 3 (March 2011): 198–211. http://dx.doi.org/10.1016/j.cagd.2011.01.003.

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33

Kitamura, Yoshifumi, Andrew Smith, Haruo Takemura, and Fumio Kishino. "A Real-Time Algorithm for Accurate Collision Detection for Deformable Polyhedral Objects." Presence: Teleoperators and Virtual Environments 7, no. 1 (February 1998): 36–52. http://dx.doi.org/10.1162/105474698565514.

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We propose an accurate collision detection algorithm for use in virtual reality applications. The algorithm works for three-dimensional graphical environments where multiple objects, represented as polyhedra (boundary representation), are undergoing arbitrary motion (translation and rotation). The algorithm can be used directly for both convex and concave objects and objects can be deformed (nonrigid) during motion. The algorithm works efficiently by first reducing the number of face pairs that need to be checked accurately for interference, by first localizing possible collision regions using bounding box and spatial subdivision techniques. Face pairs that remain after this pruning stage are then accurately checked for interference. The algorithm is efficient, simple to implement, and does not require any memory-intensive auxiliary data structures to be precomputed and updated. The performance of the proposed algorithm is compared directly against other existing algorithms, e.g., the separating plane algorithm, octree update method, and distance-based method. Results are given to show the efficiency of the proposed method in a general environment.

34

Le-Thi-Thu, Nga, Khoi Nguyen-Tan, and Thuy Nguyen-Thanh. "Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme." Journal of Science and Technology: Issue on Information and Communications Technology 3, no. 1 (March 31, 2017): 82. http://dx.doi.org/10.31130/jst.2017.41.

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Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.

35

Gillespie, Mark, Nicholas Sharp, and Keenan Crane. "Integer coordinates for intrinsic geometry processing." ACM Transactions on Graphics 40, no. 6 (December 2021): 1–13. http://dx.doi.org/10.1145/3478513.3480522.

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This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Many applications demand a correspondence between the intrinsic triangulation and the input surface, but existing data structures either rely on floating point values to encode correspondence, or do not support remeshing operations beyond basic edge flips. We instead provide an integer-based data structure that guarantees valid correspondence, even for meshes with near-degenerate elements. Our starting point is the framework of normal coordinates from geometric topology, which we extend to the broader set of operations needed for mesh processing (vertex insertion, edge splits, etc. ). The resulting data structure can be used as a drop-in replacement for earlier schemes, automatically improving reliability across a wide variety of applications. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset. In turn, we can compute reliable and highly accurate solutions to partial differential equations even on extremely low-quality meshes.

36

Boier-Martin, Ioana, and Holly Rushmeier. "Reverse Engineering Methods for Digital Restoration Applications." Journal of Computing and Information Science in Engineering 6, no. 4 (May 30, 2006): 364–71. http://dx.doi.org/10.1115/1.2356497.

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In this paper we discuss the challenges of processing and converting 3D scanned data to representations suitable for interactive manipulation in the context of virtual restoration applications. We present a constrained parametrization approach that allows us to represent 3D scanned models as parametric surfaces defined over polyhedral domains. A combination of normal- and spatial-based clustering techniques is used to generate a partition of the model into regions suitable for parametrization. Constraints can be optionally imposed to enforce a strict correspondence between input and output features. We consider two types of virtual restoration methods: (a) a paint restoration method that takes advantage of the normal-based coarse partition to identify large regions of reduced metric distortion suitable for texture mapping and (b) a shape restoration approach that relies on a refined partition used to convert the input model to a multiresolution subdivision representation suitable for intuitive interactive manipulation during digital studies of historical artifacts.

37

Chen, Xiang, Xiong Zhang, and Zupeng Jia. "A robust and efficient polyhedron subdivision and intersection algorithm for three-dimensional MMALE remapping." Journal of Computational Physics 338 (June 2017): 1–17. http://dx.doi.org/10.1016/j.jcp.2017.02.029.

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38

LANG, URS. "INJECTIVE HULLS OF CERTAIN DISCRETE METRIC SPACES AND GROUPS." Journal of Topology and Analysis 05, no. 03 (August 25, 2013): 297–331. http://dx.doi.org/10.1142/s1793525313500118.

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Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E (X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E (X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in [Formula: see text], for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space [Formula: see text] for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.

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Gurin, A. M. "A Property of the Normal Subdivision of Space into Polyhedra Induced by a Packing of Compact Bodies." Journal of Mathematical Sciences 131, no. 1 (November 2005): 5275–77. http://dx.doi.org/10.1007/s10958-005-0400-z.

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Patrikalakis, N. M., and P. V. Prakash. "Free-Form Plate Modeling Using Offset Surfaces." Journal of Offshore Mechanics and Arctic Engineering 110, no. 3 (August 1, 1988): 287–94. http://dx.doi.org/10.1115/1.3257064.

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This paper addresses the representation of plates within the framework of the Boundary Representation method in a Solid Modeling environment. Plates are defined as the volume bounded by a progenitor surface, its offset surface and other, possibly ruled surfaces for the sides. Offset surfaces of polynomial parametric surfaces cannot be represented exactly within the same class of functions describing the progenitor surface. Therefore, if the offset surface is to be represented in the same form as the progenitor surface, approximation is required. A method of approximation relevant to non-uniform rational parametric B-spline surfaces is described. The method employs the properties of the control polyhedron and a recently developed subdivision algorithm to satisfy a certain accuracy criterion. Representative examples are given which illustrate the efficiency and robustness of the proposed method.

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Isbell, RF. "Krasnozems - a profile." Soil Research 32, no. 5 (1994): 915. http://dx.doi.org/10.1071/sr9940915.

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Since the first soil map of Australia by Prescott in 1931, acid red soils developed from basalt have been specifically recognized in spite of their very limited area of occurrence in eastern Australia from North Queensland to Tasmania in a rainfall zone of about 1000 to 4000 mm. Until the early 1950s these soils were known as red loams, but the term krasnozem became formalised in 1953 with the publication of Stephen's Manual of Australian Soils. Over the past 40 years, these soils have been extensively studied because their favourable agronomic properties have led to intensive land use. The krasnozems are red to brown, acid, strongly structured clay soils (50-70% clay) ranging in depth from less than 1 m to over 7 m. Their clay mineralogy is dominated by kaolin and iron and aluminium oxides, and this ensures that the soils have variable charge properties with low cation exchange capacity and usually a significant anion exchange capacity. Free iron oxide contents range from about 7 to 18% Fe. Red basalt-derived soils occur in a number of other countries, and the 'typical' Russian krasnozems appear to have similar mineralogical and chemical properties but apparently lack the characteristic strong polyhedral structure of the Australian soils and are only about one metre deep. The Australian krasnozems are mostly classified as Oxisols in Soil Taxonomy and Ferralsols in the FAO-Unesco scheme. In the new Australian classification they are classed as Ferrosols and a more specific definition and subdivision of this class into lower categories is given, together with their relationship to morphologically similar soils.

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Eon, Jean-Guillaume. "Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings." Acta Crystallographica Section A Foundations and Advances 72, no. 3 (March 9, 2016): 268–93. http://dx.doi.org/10.1107/s2053273315022950.

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Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is thecombinatorialorbond topologyof the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal. Since periodic nets can be faithfully represented by their labelled quotient graphs, it may be inferred that their topological features can be recovered by a direct analysis of the labelled quotient graph. Evidence is given for ring analysis and structure decomposition into building units and building networks. An algebraic treatment is developed for ring analysis and thoroughly applied to a description of coesite. Building units can be finite or infinite, corresponding to 1-, 2- or even 3-periodic subnets. The list of infinite units includes linear chains or sheets of corner- or edge-sharing polyhedra. Decomposing periodic nets into their building units relies on graph-theoretical methods classified assurgery techniques. The most relevant operations are edge subdivision, vertex identification, edge contraction and decoration. Instead, these operations can be performed on labelled quotient graphs, evidencing in almost a mechanical way the nature and connection mode of building units in the derived net. Various examples are discussed, ranging from finite building blocks to 3-periodic subnets. Among others, the structures of strontium oxychloride, spinel, lithiophilite and garnet are addressed.

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OSIPENKO, GEORGE. "Symbolic images and invariant measures of dynamical systems." Ergodic Theory and Dynamical Systems 30, no. 4 (July 17, 2009): 1217–37. http://dx.doi.org/10.1017/s0143385709000431.

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AbstractLet f be a homeomorphism of a compact manifold M. The Krylov–Bogoloubov theorem guarantees the existence of a measure that is invariant with respect to f. The set of all invariant measures ℳ(f) is convex and compact in the weak topology. The goal of this paper is to construct the set ℳ(f). To obtain an approximation of ℳ(f), we use the symbolic image with respect to a partition C={M(1),M(2),…,M(n)} of M. A symbolic image G is a directed graph such that a vertex i corresponds to the cell M(i) and an edge i→j exists if and only if f(M(i))∩M(j)≠0̸. This approach lets us apply the coding of orbits and symbolic dynamics to arbitrary dynamical systems. A flow on the symbolic image is a probability distribution on the edges which satisfies Kirchhoff’s law at each vertex, i.e. the incoming flow equals the outgoing one. Such a distribution is an approximation to some invariant measure. The set of flows on the symbolic image G forms a convex polyhedron ℳ(G) which is an approximation to the set of invariant measures ℳ(f). By considering a sequence of subdivisions of the partitions, one gets sequence of symbolic images Gk and corresponding approximations ℳ(Gk) which tend to ℳ(f) as the diameter of the cells goes to zero. If the flows mk on each Gk are chosen in a special manner, then the sequence {mk} converges to some invariant measure. Every invariant measure can be obtained by this method. Applications and numerical examples are given.

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Kliem, Jonathan, and Christian Stump. "A New Face Iterator for Polyhedra and for More General Finite Locally Branched Lattices." Discrete & Computational Geometry, March 18, 2022. http://dx.doi.org/10.1007/s00454-021-00344-x.

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AbstractWe discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, García-Sánchez, O’Neill, and Wilburne, we apply this algorithm to prove Wilf’s conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf’s conjecture.

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Tao, Yujie, Chunfeng Suo, and Guijun Wang. "Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process1." Journal of Intelligent & Fuzzy Systems, September 16, 2021, 1–15. http://dx.doi.org/10.3233/jifs-210770.

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Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

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"CONSTRUCTION OF CATMULL-CLARK SUBDIVISION SCHEME." Transactions in Mathematical and Computational Sciences, December 31, 2021, 68–75. http://dx.doi.org/10.52587/tmcs0102061.

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Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

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"CONSTRUCTION OF CATMULL-CLARK SUBDIVISION SCHEME." Transactions in Mathematical and Computational Sciences, December 31, 2021, 68–77. http://dx.doi.org/10.52587/tmcs010206.

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Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

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Zhang, Jingjing, Yufeng Tian, and Xin Li. "Improved non-uniform subdivision scheme with modified Eigen-polyhedron." Visual Computing for Industry, Biomedicine, and Art 5, no. 1 (July 11, 2022). http://dx.doi.org/10.1186/s42492-022-00115-2.

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AbstractIn this study, a systematic refinement method was developed for non-uniform Catmull-Clark subdivision surfaces to improve the quality of the surface at extraordinary points (EPs). The developed method modifies the eigenpolyhedron by designing the angles between two adjacent edges that contain an EP. Refinement rules are then formulated with the help of the modified eigenpolyhedron. Numerical experiments show that the method significantly improves the performance of the subdivision surface for non-uniform parameterization.

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Arseneva, Elena, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, and Alexander Wolff. "Adjacency Graphs of Polyhedral Surfaces." Discrete & Computational Geometry, October 18, 2023. http://dx.doi.org/10.1007/s00454-023-00537-6.

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AbstractWe study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $${\mathbb {R}}^3$$ R 3 . We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $$K_5$$ K 5 , $$K_{5,81}$$ K 5 , 81 , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $$K_{4,4}$$ K 4 , 4 , and $$K_{3,5}$$ K 3 , 5 can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (Isr. J. Math. 46(1–2), 127–144 (1983)), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in $$\Omega (n\log n)$$ Ω ( n log n ) . From the non-realizability of $$K_{5,81}$$ K 5 , 81 , we obtain that any realizable n-vertex graph has $${\mathcal {O}}(n^{9/5})$$ O ( n 9 / 5 ) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

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Tanaka, Kohei. "Sectional category of maps related to finite spaces." Topological Methods in Nonlinear Analysis, March 3, 2024, 1–21. http://dx.doi.org/10.12775/tmna.2023.029.

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In this study, we compute some examples of sectional category secat$(f)$ and sectional number sec$(f) for continuous maps $f$ related to finite spaces. Moreover, we introduce an invariant secat$_k(f)$ for a map $f$ between finite spaces using the $k$-th barycentric subdivision and show the equality secat$_k(f)=$ secat$(\mathcal{B}(f))$ for sufficiently large $k$, where $\mathcal{B}(f)$ is the induced map on the associated polyhedra.