TY - GEN
T1 - A Parameterization of the Klein Bottle by Isometric Transformations in with Mathematica
AU - Velezmoro-León, Ricardo
AU - Farias-Morcillo, Nestor Javier
AU - Ipanaqué-Chero, Robert
AU - Estela-Vilela, José Manuel
AU - Jiménez-Vilcherrez, Judith Keren
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - Isometric transformations in (Figure Presented)
KW - Klein Bottle
KW - Moving frame
KW - Parameterization
UR - http://www.scopus.com/inward/record.url?scp=85115439663&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-86653-2_19
DO - 10.1007/978-3-030-86653-2_19
M3 - Conference contribution
AN - SCOPUS:85115439663
SN - 9783030866525
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 261
EP - 272
BT - Computational Science and Its Applications – ICCSA 2021 - 21st International Conference, Proceedings
A2 - Gervasi, Osvaldo
A2 - Murgante, Beniamino
A2 - Misra, Sanjay
A2 - Garau, Chiara
A2 - Blečić, Ivan
A2 - Taniar, David
A2 - Apduhan, Bernady O.
A2 - Rocha, Ana Maria
A2 - Tarantino, Eufemia
A2 - Torre, Carmelo Maria
PB - Springer Science and Business Media Deutschland GmbH
T2 - 21st International Conference on Computational Science and Its Applications, ICCSA 2021
Y2 - 13 September 2021 through 16 September 2021
ER -