This work aims to classify the families of curves obtained by the intersection of an arbitrary hyperbolic cylinder with an arbitrary torus sharing the same center, based on the number of their connected components and the number of their self-intersections points. The graphic geometric representation of these curves, in GeoGebra, and the respective algebraic descriptions, supported from a theoretical and computational point of view, are of fundamental importance for the development of this work. In this paper, we describe the procedure and the necessary implementation to achieve the outlined objective.
[1] Breda, A. M., Trocado, A., Dos Santos, J. M.: The intersection curve of an ellipsoid with a torus sharing the same center. In: Proceedings
of the 20th International Conference on Geometry and Graphics (ICGG2022), 127-137. Springer International Publishing (2023).
https://doi.org/10.1007/978-3-031-13588-0_11
[2] Breda, A. M., Trocado, A., Dos Santos, J. M.: Torus and quadrics intersection using GeoGebra. In: Proceedings of the 19th International
Conference on Geometry and Graphics (ICGG2020), 484-493. Springer International Publishing (2021). https://doi.org/10.1007/978-3-
030-63403-2_43
[3] Gonzalez-Vega, L., Trocado, A.: Using maple to compute the intersection curve of two quadrics: Improving the intersectplot command. Maple in
Mathematics Education and Research, 92-100. Springer International Publishing (2020). https://doi.org/10.1007/978-3-030-41258-6_7
[4] Gonzalez-Vega, L., Trocado, A.: Tools for analyzing the intersection curve between two quadrics through projection and lifting. Journal of
Computational and Applied Mathematics, 393, 113522 (2021). https://doi.org/10.1016/j.cam.2021.113522
[5] Kim, K., Kim, M., Oh, K.: Torus/sphere intersection based on a configuration space approach. Graphical Models and Image Processing, 60 (1),
77–92 (1998). https://doi.org/10.1006/gmip.1997.0451
[6] Pironti, A., Walker, M.: Fusion, tokamaks, and plasma control: an introduction and tutorial. IEEE Control Systems Magazine, 25 (5), 30–43
(2005). https://10.0.4.85/MCS.2005.1512794
[7] Gonzalez-Vega, L., Trocado, A., Dos Santos, J. M.: Intersecting two quadrics with GeoGebra. Algebraic Informatics, 237-248. Springer
International Publishing (2019). https://doi.org/10.1007/978-3-030-21363-3_20
[8] Gonzalez-Vega, L.: A subresultant theory for multivariate polynomials. In: Proceedings of the International Symposium on Symbolic and
Algebraic Computation (ISSAC’91), 79-85. ACM (1991).
[1] Breda, A. M., Trocado, A., Dos Santos, J. M.: The intersection curve of an ellipsoid with a torus sharing the same center. In: Proceedings
of the 20th International Conference on Geometry and Graphics (ICGG2022), 127-137. Springer International Publishing (2023).
https://doi.org/10.1007/978-3-031-13588-0_11
[2] Breda, A. M., Trocado, A., Dos Santos, J. M.: Torus and quadrics intersection using GeoGebra. In: Proceedings of the 19th International
Conference on Geometry and Graphics (ICGG2020), 484-493. Springer International Publishing (2021). https://doi.org/10.1007/978-3-
030-63403-2_43
[3] Gonzalez-Vega, L., Trocado, A.: Using maple to compute the intersection curve of two quadrics: Improving the intersectplot command. Maple in
Mathematics Education and Research, 92-100. Springer International Publishing (2020). https://doi.org/10.1007/978-3-030-41258-6_7
[4] Gonzalez-Vega, L., Trocado, A.: Tools for analyzing the intersection curve between two quadrics through projection and lifting. Journal of
Computational and Applied Mathematics, 393, 113522 (2021). https://doi.org/10.1016/j.cam.2021.113522
[5] Kim, K., Kim, M., Oh, K.: Torus/sphere intersection based on a configuration space approach. Graphical Models and Image Processing, 60 (1),
77–92 (1998). https://doi.org/10.1006/gmip.1997.0451
[6] Pironti, A., Walker, M.: Fusion, tokamaks, and plasma control: an introduction and tutorial. IEEE Control Systems Magazine, 25 (5), 30–43
(2005). https://10.0.4.85/MCS.2005.1512794
[7] Gonzalez-Vega, L., Trocado, A., Dos Santos, J. M.: Intersecting two quadrics with GeoGebra. Algebraic Informatics, 237-248. Springer
International Publishing (2019). https://doi.org/10.1007/978-3-030-21363-3_20
[8] Gonzalez-Vega, L.: A subresultant theory for multivariate polynomials. In: Proceedings of the International Symposium on Symbolic and
Algebraic Computation (ISSAC’91), 79-85. ACM (1991).
Breda, A., Trocado, A., & Dos Santos, J. M. (2024). The Intersection Curve of an Hyperbolic Cylinder with a Torus Sharing the Same Center. International Electronic Journal of Geometry, 17(2), 336-347. https://doi.org/10.36890/iejg.1318186
AMA
Breda A, Trocado A, Dos Santos JM. The Intersection Curve of an Hyperbolic Cylinder with a Torus Sharing the Same Center. Int. Electron. J. Geom. October 2024;17(2):336-347. doi:10.36890/iejg.1318186
Chicago
Breda, Ana, Alexandre Trocado, and José Manuel Dos Santos. “The Intersection Curve of an Hyperbolic Cylinder With a Torus Sharing the Same Center”. International Electronic Journal of Geometry 17, no. 2 (October 2024): 336-47. https://doi.org/10.36890/iejg.1318186.
EndNote
Breda A, Trocado A, Dos Santos JM (October 1, 2024) The Intersection Curve of an Hyperbolic Cylinder with a Torus Sharing the Same Center. International Electronic Journal of Geometry 17 2 336–347.
IEEE
A. Breda, A. Trocado, and J. M. Dos Santos, “The Intersection Curve of an Hyperbolic Cylinder with a Torus Sharing the Same Center”, Int. Electron. J. Geom., vol. 17, no. 2, pp. 336–347, 2024, doi: 10.36890/iejg.1318186.
ISNAD
Breda, Ana et al. “The Intersection Curve of an Hyperbolic Cylinder With a Torus Sharing the Same Center”. International Electronic Journal of Geometry 17/2 (October 2024), 336-347. https://doi.org/10.36890/iejg.1318186.
JAMA
Breda A, Trocado A, Dos Santos JM. The Intersection Curve of an Hyperbolic Cylinder with a Torus Sharing the Same Center. Int. Electron. J. Geom. 2024;17:336–347.
MLA
Breda, Ana et al. “The Intersection Curve of an Hyperbolic Cylinder With a Torus Sharing the Same Center”. International Electronic Journal of Geometry, vol. 17, no. 2, 2024, pp. 336-47, doi:10.36890/iejg.1318186.
Vancouver
Breda A, Trocado A, Dos Santos JM. The Intersection Curve of an Hyperbolic Cylinder with a Torus Sharing the Same Center. Int. Electron. J. Geom. 2024;17(2):336-47.