{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T16:12:38Z","timestamp":1776442358665,"version":"3.51.2"},"reference-count":11,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/00212\/2019"],"award-info":[{"award-number":["UID\/MAT\/00212\/2019"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/04561\/2019"],"award-info":[{"award-number":["UID\/MAT\/04561\/2019"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,5,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The isotropy projection establishes a correspondence between curves in the Lorentz\u2013Minkowski space\n                    <jats:inline-formula id=\"j_forum-2019-0092_ineq_9999\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msubsup>\n                            <m:mi>\ud835\udc04<\/m:mi>\n                            <m:mn>1<\/m:mn>\n                            <m:mn>3<\/m:mn>\n                          <\/m:msubsup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0092_eq_0324.png\"\/>\n                        <jats:tex-math>{\\mathbf{E}_{1}^{3}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane\n                    <jats:inline-formula id=\"j_forum-2019-0092_ineq_9998\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msup>\n                            <m:mi mathvariant=\"script\">\u2112<\/m:mi>\n                            <m:mn>2<\/m:mn>\n                          <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0092_eq_0355.png\"\/>\n                        <jats:tex-math>{\\mathcal{L}^{2}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ). In this paper, we shall\ngive necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in\n                    <jats:inline-formula id=\"j_forum-2019-0092_ineq_9997\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msubsup>\n                            <m:mi>\ud835\udc04<\/m:mi>\n                            <m:mn>1<\/m:mn>\n                            <m:mn>3<\/m:mn>\n                          <\/m:msubsup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0092_eq_0324.png\"\/>\n                        <jats:tex-math>{\\mathbf{E}_{1}^{3}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We shall discuss the causal character of the second derivative of unit speed spacelike curves in\n                    <jats:inline-formula id=\"j_forum-2019-0092_ineq_9996\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msubsup>\n                            <m:mi>\ud835\udc04<\/m:mi>\n                            <m:mn>1<\/m:mn>\n                            <m:mn>3<\/m:mn>\n                          <\/m:msubsup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_forum-2019-0092_eq_0324.png\"\/>\n                        <jats:tex-math>{\\mathbf{E}_{1}^{3}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.\n                  <\/jats:p>","DOI":"10.1515\/forum-2019-0092","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T04:01:53Z","timestamp":1581393713000},"page":"693-711","source":"Crossref","is-referenced-by-count":1,"title":["Envelopes of circles and spacelike curves in the Lorentz\u2013Minkowski 3-space"],"prefix":"10.1515","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9578-2380","authenticated-orcid":false,"given":"Rui","family":"Pacheco","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI) , Universidade da Beira Interior , 6201-001 Covilh\u00e3 , Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7817-3989","authenticated-orcid":false,"given":"Susana D.","family":"Santos","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica, Aplica\u00e7\u00f5es Fundamentais e Investiga\u00e7\u00e3o Operacional and Departamento de Matem\u00e1tica , Faculdade de Ci\u00eancias , Universidade de Lisboa , 1749-016 Lisboa , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2020,2,11]]},"reference":[{"key":"2026041715203610890_j_forum-2019-0092_ref_001","doi-asserted-by":"crossref","unstructured":"W.  Blaschke,\nVorlesungen \u00fcber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativit\u00e4tstheorie. Vol. 3,\nSpringer, Berlin, 1929.","DOI":"10.1007\/978-3-642-50823-3"},{"key":"2026041715203610890_j_forum-2019-0092_ref_002","doi-asserted-by":"crossref","unstructured":"T. E.  Cecil,\nLie Sphere Geometry,\nUniversitext,\nSpringer, New York, 1992.","DOI":"10.1007\/978-1-4757-4096-7"},{"key":"2026041715203610890_j_forum-2019-0092_ref_003","doi-asserted-by":"crossref","unstructured":"J. P.  Fillmore and A.  Springer,\nNew Euclidean theorems by the use of Laguerre transformations \u2013 Some geometry of Minkowski (2+1)-space,\nJ. Geom. 52 (1995), no. 1\u20132, 74\u201390.","DOI":"10.1007\/BF01406828"},{"key":"2026041715203610890_j_forum-2019-0092_ref_004","unstructured":"F.  Gomes Teixeira,\nTratado de las curvas especiales notables,\nAcademia de Ciencias, Madrid, 1905."},{"key":"2026041715203610890_j_forum-2019-0092_ref_005","doi-asserted-by":"crossref","unstructured":"T.  Kubota,\nOn the differential invariants of the Laguerre group,\nProc. Camb. Philos. Soc. 22 (1924), no. 2, 113\u2013123.","DOI":"10.1017\/S0305004100002851"},{"key":"2026041715203610890_j_forum-2019-0092_ref_006","doi-asserted-by":"crossref","unstructured":"R.  L\u00f3pez,\nDifferential geometry of curves and surfaces in Lorentz\u2013Minkowski space,\nInt. Electron. J. Geom. 7 (2014), no. 1, 44\u2013107.","DOI":"10.36890\/iejg.594497"},{"key":"2026041715203610890_j_forum-2019-0092_ref_007","unstructured":"E.  Musso and L.  Nicolodi,\nSurfaces in Laguerre geometry,\npreprint (2018), https:\/\/arxiv.org\/abs\/1802.05507."},{"key":"2026041715203610890_j_forum-2019-0092_ref_008","doi-asserted-by":"crossref","unstructured":"B.  Nolasco and R.  Pacheco,\nEvolutes of plane curves and null curves in Minkowski 3-space,\nJ. Geom. 108 (2017), no. 1, 195\u2013214.","DOI":"10.1007\/s00022-016-0334-2"},{"key":"2026041715203610890_j_forum-2019-0092_ref_009","doi-asserted-by":"crossref","unstructured":"H.  Pottmann and M.  Peternell,\nApplications of Laguerre geometry in CAGD,\nComput. Aided Geom. Design 15 (1998), no. 2, 165\u2013186.","DOI":"10.1016\/S0167-8396(97)00023-X"},{"key":"2026041715203610890_j_forum-2019-0092_ref_010","doi-asserted-by":"crossref","unstructured":"M.  Skopenkov, H.  Pottmann and P.  Grohs,\nRuled Laguerre minimal surfaces,\nMath. Z. 272 (2012), no. 1\u20132, 645\u2013674.","DOI":"10.1007\/s00209-011-0953-0"},{"key":"2026041715203610890_j_forum-2019-0092_ref_011","unstructured":"H.  Stachel,\nStrophoids: A familiy of cubic curves with remarkable properties,\nJ. Indust. Design Eng. Graphics 10 (2015), 65\u201372."}],"container-title":["Forum Mathematicum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/form\/32\/3\/article-p693.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/forum-2019-0092\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/forum-2019-0092\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T15:20:48Z","timestamp":1776439248000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/forum-2019-0092\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,11]]},"references-count":11,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,2,11]]},"published-print":{"date-parts":[[2020,5,1]]}},"alternative-id":["10.1515\/forum-2019-0092"],"URL":"https:\/\/doi.org\/10.1515\/forum-2019-0092","relation":{},"ISSN":["0933-7741","1435-5337"],"issn-type":[{"value":"0933-7741","type":"print"},{"value":"1435-5337","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,11]]}}}