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首页> 外文期刊>International Journal of Mathematical Education in Science and Technology >A complete description of cones and polytopes including hypervolumes of all facets of a polytope
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A complete description of cones and polytopes including hypervolumes of all facets of a polytope

机译：圆锥体和多面体的完整描述，包括多面体所有小面的超体积

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In this paper methods and algorithms for identifying the main elements (edges and facets of any dimension) of a cone and a polytope, and calculating the corresponding hypervolumes are presented. The cones and polytopes are supposed to be given as the non-negative linear combination and the convex hull generated by a, not necessarily minimal, set of vectors (points), respectively, and they can be degenerated (of a dimension smaller that that of the proper space in which they are contained). First a minimum set of generators (edges and vertices) are obtained by eliminating the redundant vectors. In the case of cones, the linear space basis and the minimal cone generators are obtained. Second the set of all facets of any dimension are identified. Finally, an algorithm for obtaining the associated hypervolumes of any dimension, i.e. the length of its edges, the areas of its faces of dimension two, and the hypervolumes of its facets of any dimension, is introduced. The proposed formula leads to a recursion that gives the hypervolumes of dimension n as a function of other hypervolumes of dimension n - 1. Examples are used to illustrate the proposed methods and algorithms.
机译：在本文中，提出了用于识别圆锥体和多面体的主要元素（任意尺寸的边和小面）并计算相应的超体积的方法和算法。假定视锥和多面体分别是非负线性组合和凸包，它们分别由一组（不一定是最小的）向量（点）生成，并且它们可以退化（维数小于维）。包含它们的适当空间）。首先，通过消除冗余向量获得最小的一组生成器（边和顶点）。对于圆锥体，可以获得线性空间基础和最小圆锥体生成器。其次，确定任何维度的所有方面的集合。最后，介绍了一种用于获得任何尺寸的相关超体积的算法，即，其边缘的长度，其尺寸为2的面的面积以及其任何尺寸的面的超体积。所提出的公式导致一个递归，该递归给出了维度n的超体积与维度n-1的其他超体积的函数。示例用于说明所提出的方法和算法。

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