Partial Covering of a Circle by 6 and 7 Congruent Circles
Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by <i>n</i> congruent circles of given radius <i>r</i>, while <i>r</i> varies from the radius in the maximum packing to the radius...
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| Autores principales: | , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/edbee9e084774953989571de9d919abb |
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| Sumario: | Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by <i>n</i> congruent circles of given radius <i>r</i>, while <i>r</i> varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for <i>n</i> = 2 to 5. In the present paper, numerical solutions are given to this problem for <i>n</i> = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of <i>n</i> = 6 is very complicated, whilst the case <i>n</i> = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate. |
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