{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:45Z","timestamp":1760241885874,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,2]],"date-time":"2018-10-02T00:00:00Z","timestamp":1538438400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work describes the geometry behind the Smith chart, recent 3D Smith chart tool and previously reported conceptual Hyperbolic Smith chart. We present the geometrical properties of the transformations used in creating them by means of inversive geometry and basic non-Euclidean geometry. The beauty and simplicity of this perspective are complementary to the classical way in which the Smith chart is taught in the electrical engineering community by providing a visual insight that can lead to new developments. Further we extend our previous work where we have just drawn the conceptual hyperbolic Smith chart by providing the equations for its generation and introducing additional properties.<\/jats:p>","DOI":"10.3390\/sym10100458","type":"journal-article","created":{"date-parts":[[2018,10,2]],"date-time":"2018-10-02T11:30:02Z","timestamp":1538479802000},"page":"458","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0403-1405","authenticated-orcid":false,"given":"Mar\u00eda Jose","family":"P\u00e9rez-Pe\u00f1alver","sequence":"first","affiliation":[{"name":"Universitat Polit\u00e8cnica de Val\u00e8ncia, Matem\u00e1tica Aplicada, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4523-1991","authenticated-orcid":false,"given":"Esther","family":"Sanabria-Codesal","sequence":"additional","affiliation":[{"name":"Universitat Polit\u00e8cnica de Val\u00e8ncia, Matem\u00e1tica Aplicada, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Florica","family":"Moldoveanu","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Faculty of Automatic Control and Computers, University POLITEHNICA of Bucharest, 060042 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1368-7249","authenticated-orcid":false,"given":"Alin","family":"Moldoveanu","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Faculty of Automatic Control and Computers, University POLITEHNICA of Bucharest, 060042 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4776-2542","authenticated-orcid":false,"given":"Victor","family":"Asavei","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Faculty of Automatic Control and Computers, University POLITEHNICA of Bucharest, 060042 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2613-0917","authenticated-orcid":false,"given":"Andrei","family":"A. Muller","sequence":"additional","affiliation":[{"name":"Ecole Polytechnique Federale de Lausanne, Nanoelectronic Devices Laboratory, 1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adrian","family":"Ionescu","sequence":"additional","affiliation":[{"name":"Ecole Polytechnique Federale de Lausanne, Nanoelectronic Devices Laboratory, 1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,2]]},"reference":[{"key":"ref_1","first-page":"29","article-title":"Transmission-line calculator","volume":"12","author":"Smith","year":"1939","journal-title":"Electronics"},{"key":"ref_2","unstructured":"Smith, P.H. (1969). 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