{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:38:06Z","timestamp":1760146686979,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,3]],"date-time":"2024-12-03T00:00:00Z","timestamp":1733184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["UIDB\/04106\/2020","UIDP\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020","UIDP\/04106\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The parabolic cylinders under analysis have generatrices parallel to the axis of the torus, whereas the elliptical cylinders, centered at the same point as the torus, have axes either aligned with or orthogonal to the torus\u2019s axis. For the topological classification of these intersection curves, we consider their number of connected components and self-intersection points. GeoGebra, which was used to create the 3D visual geometric representations of the intersection curves, and Maple, which was used to perform the essential symbolic algebraic calculations, were critical computational tools in the development of this work. Theoretical and computational approaches are interwoven throughout this study, with the computational work serving as the foundation for exploration and providing insights that contributed to the theoretical validation of the results revealed through GeoGebra simulations.<\/jats:p>","DOI":"10.3390\/axioms13120852","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T10:07:10Z","timestamp":1733306830000},"page":"852","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Topological Properties of the Intersection Curves Between a Torus and Families of Parabolic or Elliptical Cylinders"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7076-707X","authenticated-orcid":false,"given":"Ana","family":"Breda","sequence":"first","affiliation":[{"name":"Center for Research & Development in Mathematics and Applications (CIDMA), Universidade de Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5589-8100","authenticated-orcid":false,"given":"Alexandre","family":"Trocado","sequence":"additional","affiliation":[{"name":"Center for Research & Development in Mathematics and Applications (CIDMA), Universidade de Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6830-6503","authenticated-orcid":false,"given":"Jos\u00e9","family":"Dos Santos","sequence":"additional","affiliation":[{"name":"Center for Research & Development in Mathematics and Applications (CIDMA), Universidade de Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,3]]},"reference":[{"key":"ref_1","unstructured":"Tu, C., Wang, W., and Wang, J. (2002, January 10\u201312). Classifying the nonsingular intersection curve of two quadric surfaces. Proceedings of the Geometric Modeling and Processing. Theory and Applications, GMP 2002, Wako, Japan. GMAP-02."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1109\/MCG.1983.263273","article-title":"Intersection of Parametric Surfaces by Means of Look-Up Tables","volume":"3","author":"Hanna","year":"1983","journal-title":"IEEE Comput. Graph. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/MCS.2005.1512794","article-title":"Fusion, tokamaks, and plasma control: An introduction and tutorial","volume":"25","author":"Pironti","year":"2005","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1006\/gmip.1997.0451","article-title":"Torus\/Sphere Intersection Based on a Configuration Space Approach","volume":"60","author":"Kim","year":"1998","journal-title":"Graph. Model. Image Process."},{"key":"ref_5","unstructured":"Kim, M.S. (1999, January 1\u20134). Intersecting surfaces of special types. Proceedings of the Shape Modeling International \u201999. International Conference on Shape Modeling and Applications, Aizu-Wakamatsu, Japan."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2387","DOI":"10.1016\/j.cam.2011.11.025","article-title":"Circles in torus\u2013torus intersections","volume":"236","author":"Kim","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"465","DOI":"10.3722\/cadaps.2011.465-477","article-title":"Torus\/Torus Intersection","volume":"8","author":"Liu","year":"2011","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1080\/16864360.2004.10738288","article-title":"Algebraic Algorithms for Computing Intersections between Torus and Natural Quadrics","volume":"1","author":"Li","year":"2004","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"113522","DOI":"10.1016\/j.cam.2021.113522","article-title":"Tools for analyzing the intersection curve between two quadrics through projection and lifting","volume":"393","author":"Trocado","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Dupont, L., Lazard, D., Lazard, S., and Petitjean, S. (2003, January 8\u201310). Near-optimal parameterization of the intersection of quadrics. Proceedings of the SCG \u201903, San Diego, CA, USA. Available online: https:\/\/api.semanticscholar.org\/CorpusID:26759864.","DOI":"10.1145\/777829.777830"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"\u0106iri\u0107, M., Droste, M., and Pin, J.\u00c9. (2019). Intersecting Two Quadrics with GeoGebra. Algebraic Informatics, Springer.","DOI":"10.1007\/978-3-030-21363-3"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Gerhard, J., and Kotsireas, I. (2020). Using Maple to Compute the Intersection Curve of Two Quadrics: Improving the Intersectplot Command. Maple in Mathematics Education and Research, Springer.","DOI":"10.1007\/978-3-030-41258-6"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Cheng, L.Y. (2021). Torus and Quadrics Intersection Using GeoGebra. ICGG 2020\u2014Proceedings of the 19th International Conference on Geometry and Graphics, Springer.","DOI":"10.1007\/978-3-030-63403-2"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/852\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:45:53Z","timestamp":1760114753000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/852"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,3]]},"references-count":13,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["axioms13120852"],"URL":"https:\/\/doi.org\/10.3390\/axioms13120852","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,3]]}}}