Billiards in ellipses revisited

Billiards in ellipses revisited

We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proof...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:European journal of mathematics 2022-12, Vol.8 (4), p.1313-1327
Hauptverfasser: Akopyan, Arseniy, Schwartz, Richard, Tabachnikov, Serge
Format: Artikel
Sprache:eng
Schlagworte:
Algebraic Geometry
Angles (geometry)
Ellipses
Mathematics
Mathematics and Statistics
Polygons
Research Article
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 1327
container_issue 4
container_start_page 1313
container_title European journal of mathematics
container_volume 8
creator Akopyan, Arseniy
Schwartz, Richard
Tabachnikov, Serge
description We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.
doi_str_mv 10.1007/s40879-020-00426-9
format Article
fullrecord <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2736746234</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2736746234</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-2be409fb2e3671242a3898cacccbbbf78ae862697c78d8a6a807e01ad0ad48e63</originalsourceid><addsrcrecordid>eNp9kEtLxDAUhYMoOIzzB9xYcB29eZjHUgcdhQE3Cu5Cmt5KpLY16Qj-e6MV3bm6Z3G-c-Ej5JjBGQPQ51mC0ZYCBwoguaJ2jyw4s5Yqrcz-b754OiSrnGMNgnElBJMLcnIVuy761OQq9hWWPGbMVcL3mOOEzRE5aH2XcfVzl-Tx5vphfUu395u79eWWBsHsRHmNEmxbcxRKMy65F8aa4EMIdV232ng0iiurgzaN8cob0AjMN-AbaVCJJTmdd8c0vO0wT-5l2KW-vHRcl02puJClxedWSEPOCVs3pvjq04dj4L5cuNmFKy7ctwtnCyRmKJdy_4zpb_of6hO_8GBt</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2736746234</pqid></control><display><type>article</type><title>Billiards in ellipses revisited</title><source>SpringerLink Journals - AutoHoldings</source><creator>Akopyan, Arseniy ; Schwartz, Richard ; Tabachnikov, Serge</creator><creatorcontrib>Akopyan, Arseniy ; Schwartz, Richard ; Tabachnikov, Serge</creatorcontrib><description>We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.</description><identifier>ISSN: 2199-675X</identifier><identifier>EISSN: 2199-6768</identifier><identifier>DOI: 10.1007/s40879-020-00426-9</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Algebraic Geometry ; Angles (geometry) ; Ellipses ; Mathematics ; Mathematics and Statistics ; Polygons ; Research Article</subject><ispartof>European journal of mathematics, 2022-12, Vol.8 (4), p.1313-1327</ispartof><rights>Springer Nature Switzerland AG 2020</rights><rights>Springer Nature Switzerland AG 2020.</rights><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-2be409fb2e3671242a3898cacccbbbf78ae862697c78d8a6a807e01ad0ad48e63</citedby><cites>FETCH-LOGICAL-c319t-2be409fb2e3671242a3898cacccbbbf78ae862697c78d8a6a807e01ad0ad48e63</cites><orcidid>0000-0001-5398-7820</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s40879-020-00426-9$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40879-020-00426-9$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Akopyan, Arseniy</creatorcontrib><creatorcontrib>Schwartz, Richard</creatorcontrib><creatorcontrib>Tabachnikov, Serge</creatorcontrib><title>Billiards in ellipses revisited</title><title>European journal of mathematics</title><addtitle>European Journal of Mathematics</addtitle><description>We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.</description><subject>Algebraic Geometry</subject><subject>Angles (geometry)</subject><subject>Ellipses</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polygons</subject><subject>Research Article</subject><issn>2199-675X</issn><issn>2199-6768</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAUhYMoOIzzB9xYcB29eZjHUgcdhQE3Cu5Cmt5KpLY16Qj-e6MV3bm6Z3G-c-Ej5JjBGQPQ51mC0ZYCBwoguaJ2jyw4s5Yqrcz-b754OiSrnGMNgnElBJMLcnIVuy761OQq9hWWPGbMVcL3mOOEzRE5aH2XcfVzl-Tx5vphfUu395u79eWWBsHsRHmNEmxbcxRKMy65F8aa4EMIdV232ng0iiurgzaN8cob0AjMN-AbaVCJJTmdd8c0vO0wT-5l2KW-vHRcl02puJClxedWSEPOCVs3pvjq04dj4L5cuNmFKy7ctwtnCyRmKJdy_4zpb_of6hO_8GBt</recordid><startdate>20221201</startdate><enddate>20221201</enddate><creator>Akopyan, Arseniy</creator><creator>Schwartz, Richard</creator><creator>Tabachnikov, Serge</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-5398-7820</orcidid></search><sort><creationdate>20221201</creationdate><title>Billiards in ellipses revisited</title><author>Akopyan, Arseniy ; Schwartz, Richard ; Tabachnikov, Serge</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-2be409fb2e3671242a3898cacccbbbf78ae862697c78d8a6a807e01ad0ad48e63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebraic Geometry</topic><topic>Angles (geometry)</topic><topic>Ellipses</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polygons</topic><topic>Research Article</topic><toplevel>online_resources</toplevel><creatorcontrib>Akopyan, Arseniy</creatorcontrib><creatorcontrib>Schwartz, Richard</creatorcontrib><creatorcontrib>Tabachnikov, Serge</creatorcontrib><collection>CrossRef</collection><jtitle>European journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Akopyan, Arseniy</au><au>Schwartz, Richard</au><au>Tabachnikov, Serge</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Billiards in ellipses revisited</atitle><jtitle>European journal of mathematics</jtitle><stitle>European Journal of Mathematics</stitle><date>2022-12-01</date><risdate>2022</risdate><volume>8</volume><issue>4</issue><spage>1313</spage><epage>1327</epage><pages>1313-1327</pages><issn>2199-675X</issn><eissn>2199-6768</eissn><abstract>We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s40879-020-00426-9</doi><tpages>15</tpages><orcidid>https://orcid.org/0000-0001-5398-7820</orcidid></addata></record>
fulltext fulltext
identifier ISSN: 2199-675X
ispartof European journal of mathematics, 2022-12, Vol.8 (4), p.1313-1327
issn 2199-675X
2199-6768
language eng
recordid cdi_proquest_journals_2736746234
source SpringerLink Journals - AutoHoldings
subjects Algebraic Geometry
Angles (geometry)
Ellipses
Mathematics
Mathematics and Statistics
Polygons
Research Article
title Billiards in ellipses revisited
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-06T07%3A22%3A14IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Billiards%20in%20ellipses%20revisited&rft.jtitle=European%20journal%20of%20mathematics&rft.au=Akopyan,%20Arseniy&rft.date=2022-12-01&rft.volume=8&rft.issue=4&rft.spage=1313&rft.epage=1327&rft.pages=1313-1327&rft.issn=2199-675X&rft.eissn=2199-6768&rft_id=info:doi/10.1007/s40879-020-00426-9&rft_dat=%3Cproquest_cross%3E2736746234%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2736746234&rft_id=info:pmid/&rfr_iscdi=true