Generalization of the Apollonius theorem for simplices and related problems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911749432672256 |
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| author | Vrahatis, Michael N. |
| author_facet | Vrahatis, Michael N. |
| contents | The Apollonius theorem gives the length of a median of a triangle in terms of the lengths of its sides. The straightforward generalization of this theorem obtained for m-simplices in the n-dimensional Euclidean space for n greater than or equal to m is given. Based on this, generalizations of properties related to the medians of a triangle are presented. In addition, applications of the generalized Apollonius' theorem and the related to the medians results, are given for obtaining: (a) the minimal spherical surface that encloses a given simplex or a given bounded set, (b) the thickness of a simplex that it provides a measure for the quality or how well shaped a simplex is, and (c) the convergence and error estimates of the root-finding bisection method applied on simplices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03232 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalization of the Apollonius theorem for simplices and related problems Vrahatis, Michael N. Geometric Topology Computational Geometry Numerical Analysis Primary 51M05, 52A40. Secondary 52A35, 53A07, 51M20, 51M25, 52A37 The Apollonius theorem gives the length of a median of a triangle in terms of the lengths of its sides. The straightforward generalization of this theorem obtained for m-simplices in the n-dimensional Euclidean space for n greater than or equal to m is given. Based on this, generalizations of properties related to the medians of a triangle are presented. In addition, applications of the generalized Apollonius' theorem and the related to the medians results, are given for obtaining: (a) the minimal spherical surface that encloses a given simplex or a given bounded set, (b) the thickness of a simplex that it provides a measure for the quality or how well shaped a simplex is, and (c) the convergence and error estimates of the root-finding bisection method applied on simplices. |
| title | Generalization of the Apollonius theorem for simplices and related problems |
| topic | Geometric Topology Computational Geometry Numerical Analysis Primary 51M05, 52A40. Secondary 52A35, 53A07, 51M20, 51M25, 52A37 |
| url | https://arxiv.org/abs/2401.03232 |