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On a discrete Version of Alexandrov's Projection Theorem

Acta Mathematica Sinica,English Series 2013年第8期29卷 1597-1606页

作者： Huan XIONG School of Mathematical Sciences Peking University

In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.

关键词： Aleksandrov's projection theorem convex lattice set discrete geometry

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discrete Plane Segmentation and Estimation from a Point Cloud Using Local Geometric Patterns

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International Journal of Automation and computing 2008年第3期5卷 246-256页

作者： Yukiko Kenmochi Lilian Buzer Akihiro Sugimoto Ikuko Shimizu Université Paris-Est Laboratoire d'Informatique de l'Institut Gaspard-MongeUMR CNRS 8049 A2SI-ESIEECité DescartesBP 9993162 Noisy-le-Grand CedexFrance Department of Computer Information and Communication SciencesTokyo University of Agriculture and Technology

This paper presents a method for segmenting a 3D point cloud into planar surfaces using recently obtained discretegeometry results. In discrete geometry, a discrete plane is defined as a set of grid points lying between two parallel planes with a small distance, called thickness. In contrast to the continuous case, there exist a finite number of local geometric patterns （LGPs） appearing on discrete planes. Moreover, such an LGP does not possess the unique normal vector but a set of normal vectors. By using those LGP properties, we first reject non-linear points from a point cloud, and then classify non-rejected points whose LGPs have common normal vectors into a planar-surface-point set. From each segmented point set, we also estimate the values of parameters of a discrete plane by minimizing its thickness.

关键词： discrete plane image segmentation parameter estimation discrete geometry local geometric pattern (LGP)

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On Fuzzy Simplex and Fuzzy Convex Hull

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Journal of Mathematical Research and Exposition 2011年第5期31卷 845-854页

作者： Dong QIU Wei Quan ZHANG College of Mathematics and Physics Chongqing University of Posts and Telecommunications Chongqing 400065 P. R. China Department of Mathematics School of Information Engineering Guangdong Medical College Guangdong 523808 P. R. China

In this paper, we discuss fuzzy simplex and fuzzy convex hull, and give several representation theorems for fuzzy simplex and fuzzy convex hull. In addition, by giving a new characterization theorem of fuzzy convex hu... 详细信息

关键词： discrete geometry fuzzy simplex fuzzy convex hull fuzzy geometry.

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凸F-多边形内部的F-点数

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Chinese Quarterly Journal of Mathematics 2024年第1期39卷 46-58页

作者：郭自欢魏祥林 Shijiazhuang No.1 Middle School Shijiazhuang 050000China College of Science Hebei University of Science and TechnologyShijiazhuang 050018China

An F-polygon is a simple polygon whose vertices are F-points, which are points of the set of vertices of a tiling of R~2 by regular triangles and regular hexagons of unit edge. Let f(v) denote the least possible number of F-points in the interior of a convex F-polygon K with v vertices. In this paper we prove that f(10) = 10, f(11) = 12,f(12) = 12.

关键词： discrete geometry F-polygon Interior hull Lattice polygon

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Rigidity of the Hexagonal Delaunay Triangulated Plane

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Peking Mathematical Journal 2022年第1期5卷 1-20页

作者： Song Dai Huabin Ge Shiguang Ma Center for Applied Mathematics Tianjin UniversityTianjin 300072China School of Mathematics Renmin University of ChinaBeijing 100872China School of Mathematical Sciences and LPMC Nankai UniversityTianjin 300071China

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo’s PL *** a consequence,we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

关键词： Delaunay triangulation PL conformal metric discrete geometry

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Separating Space and Time for Dimensional Analysis and Euclidean Relational Modeling

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Journal of Applied Mathematics and Physics 2023年第9期11卷 2704-2738页

作者： Steven D. P. Moore Independent Researcher Milton Canada

The theory of relativity links space and time to account for observed events in four-dimensional space. In this article we describe an alternative static state causal discrete time modeling system using an omniscient viewpoint of dynamical systems that can express object relations in the moment(s) they are observed. To do this, three key components are required, including the introduction of independent object-relative dimensional metrics, a zero-dimensional frame of reference, and application of Euclidean geometry for modeling. Procedures separate planes of matter, extensions of space (relational distance) and time (duration) using object-oriented dimensional quantities. Quantities are converted into base units using symmetry for space (Dihedral360), time (Dihedral12), rotation (Dihedral24), and scale (Dihedral10). Geometric elements construct static state outputs in discrete time models rather than continuous time using calculus, thereby using dimensional and positional natural number numerals that can visually encode complex data instead of using abstraction and irrationals. Static state Euclidean geometric models of object relations are both measured and expressed in the state they are observed in zero-time as defined by a signal. The frame can include multiple observer frames of reference where each origin, point, is the location of a distinct privileged point of reference. Two broad and diverse applications are presented: a one-dimensional spatiotemporal orbital model, and a thought experiment related to a physical theory beyond Planck limits. We suggest that expanding methodologies and continued formalization, novel tools for physics can be considered along with applications for computational discrete geometric modeling.

关键词： Spacetime Relationalism Quantum Classical Signal discrete geometry

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