Abstract
The Klein bottle plays a crucial role in the main modern sciences. This surface was first described in 1882 by the German mathematician Felix Klein. In this paper we describe a technique to obtain the parameterization of the Klein bottle. This technique uses isometric transformations (translations and rotations) and the moving frame associated with the unit circumference lying on the xy-plane. The process we follow is to start with the parametrization of the Euclidean cylinder, then continue with the parameterization of the Möbius strip, after that with the parameterization of the torus of revolution and finally, in a natural way, we describe the aforementioned technique. With the parameterization of the Klien bottle obtained, it is easy to show that it can be obtained by gluing two Möbius strips. Additionally, the parameterizations of the n-twisted and n-turns Klein bottles are obtained. All geometric calculations and geometric interpretations are performed with the Mathematica symbolic calculus system.
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Acknowledgements
The authors would like to thank to the authorities of the Universidad Nacional de Piura for the acquisition of the Mathematica 11.0 license and the reviewers for their valuable comments and suggestions.
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Velezmoro-León, R., Farias-Morcillo, N.J., Ipanaqué-Chero, R., Estela-Vilela, J.M., Jiménez-Vilcherrez, J.K. (2021). A Parameterization of the Klein Bottle by Isometric Transformations in
with Mathematica.
In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_19
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DOI: https://doi.org/10.1007/978-3-030-86653-2_19
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