空间几何单元-准多面体

A. Efremov, T. Vereschagina, N. Kadykova, V. Rustamyan
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引用次数: 4

摘要

三维空间的平铺是一种非常有趣但尚未被充分探索的平铺类型。通过凸多面体进行平铺已经得到了部分研究,例如,作品[1,15,20]致力于通过各种四面体进行平铺,一旦柏拉图,阿基米德和加泰罗尼亚体实现了平铺。瓷砖的使用从古代开始,在平面上与拼花地板和装饰品的创造有关,在空间上-与房屋的建造有关,但即使是现在,瓷砖应用的新领域也在不断开拓,例如,最近关于使用瓷砖包装信息的工作周期[17]。到目前为止,太空中的平铺几乎总是由多面体来考虑。以曲面为界的物体很少被考虑,人们可以回想起[3]中描述的双曲抛物面为界的准多面体[14]、二面体、椭球体、双圆锥体、球体[21]、Steinmetz图[22]、类地椭球体和[6]中提到的双曲四面体、立方体、八面体,以及具有有界曲面的平铺体实际上没有被考虑。除了无限的三维Schwartz曲面,但它们也被认为是曲面,而不是物体。当然,在每个这样的曲面中,您可以选择一个基本单元并用一个主体填充它,从而得到一个几何单元。通过这项工作,我们试图消除这种差距,并描述了识别由曲面分隔区隔开的几何细胞的方法。提出了紧密封装框架的概念,并描述了一种识别它们的方法。描述了一种识别多面体和准多面体几何单元的图形算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Spatial Geometric Cells — Quasipolyhedra
Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.
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