A Complete Description of Cones and Polytopes Including Hypervolumes of All Facets of a Polytope
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| Title: | A Complete Description of Cones and Polytopes Including Hypervolumes of All Facets of a Polytope |
|---|---|
| Language: | English |
| Authors: | Jubete, F., Castillo, E. |
| Source: | International Journal of Mathematical Education in Science & Technology. Jan 2007 38(1):85-102. |
| Availability: | Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals/default.html |
| Peer Reviewed: | Y |
| Page Count: | 18 |
| Publication Date: | 2007 |
| Document Type: | Journal Articles Reports - Descriptive |
| Descriptors: | Algebra, Geometric Concepts, Mathematical Formulas, Mathematical Logic, Background, Illustrations, Scientific Methodology, Mathematics Education |
| ISSN: | 0020-739X |
| Abstract: | 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 than 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. (Contains 4 tables and 1 figure.) |
| Abstractor: | Author |
| Number of References: | 22 |
| Entry Date: | 2007 |
| Access URL: | https://taylorandfrancis.metapress.com/link.asp?id=R4K5863536K1UK80 |
| Accession Number: | EJ753951 |
| Database: | ERIC |
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| Abstract: | 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 than 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. (Contains 4 tables and 1 figure.) |
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| ISSN: | 0020-739X |
