{"id":"https://openalex.org/W2963351908","doi":"https://doi.org/10.1007/s40315-018-0257-z","title":"The Ptolemy\u2013Alhazen Problem and Spherical Mirror Reflection","display_name":"The Ptolemy\u2013Alhazen Problem and Spherical Mirror Reflection","publication_year":2018,"publication_date":"2018-12-15","ids":{"openalex":"https://openalex.org/W2963351908","doi":"https://doi.org/10.1007/s40315-018-0257-z","mag":"2963351908"},"language":"en","primary_location":{"id":"doi:10.1007/s40315-018-0257-z","doi":"https://doi.org/10.1007/s40315-018-0257-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s40315-018-0257-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s40315-018-0257-z.pdf","source":{"id":"https://openalex.org/S2483979929","display_name":"Computational Methods and Function Theory","issn_l":"1617-9447","issn":["1617-9447","2195-3724"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods and Function Theory","raw_type":"journal-article"},"parent_works":[],"lineage":["https://openalex.org/W2963351908"],"child_works_count":0,"type":"article","is_traditional":true,"indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s40315-018-0257-z.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051561183","display_name":"Masayo Fujimura","orcid":"https://orcid.org/0000-0002-5837-8167","observed_orcids":["https://orcid.org/0000-0002-5837-8167"]},"institutions":[{"id":"https://openalex.org/I443798","display_name":"National Defense Academy of Japan","ror":"https://ror.org/05xszy717","country_code":"JP","type":"education","lineage":["https://openalex.org/I443798"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Masayo Fujimura","raw_affiliation_strings":["Department of Mathematics, National Defense Academy of Japan, Yokosuka, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, National Defense Academy of Japan, Yokosuka, Japan","institution_ids":["https://openalex.org/I443798"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089699151","display_name":"Parisa Hariri","orcid":"https://orcid.org/0000-0001-9849-2186","observed_orcids":["https://orcid.org/0000-0001-9849-2186"]},"institutions":[{"id":"https://openalex.org/I155660961","display_name":"University of Turku","ror":"https://ror.org/05vghhr25","country_code":"FI","type":"education","lineage":["https://openalex.org/I155660961"]}],"countries":["FI"],"is_corresponding":true,"raw_author_name":"Parisa Hariri","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Turku, Turku, Finland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Turku, Turku, Finland","institution_ids":["https://openalex.org/I155660961"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059844108","display_name":"Marcelina Mocanu","orcid":"https://orcid.org/0000-0002-0192-8180","observed_orcids":["https://orcid.org/0000-0002-0192-8180"]},"institutions":[{"id":"https://openalex.org/I4210142575","display_name":"University of Bac\u0103u","ror":"https://ror.org/03x3axr33","country_code":"RO","type":"education","lineage":["https://openalex.org/I4210142575"]}],"countries":["RO"],"is_corresponding":false,"raw_author_name":"Marcelina Mocanu","raw_affiliation_strings":["Department of Mathematics and Informatics, Vasile Alecsandri University of Bacau, Bac\u0103u, Romania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Informatics, Vasile Alecsandri University of Bacau, Bac\u0103u, Romania","institution_ids":["https://openalex.org/I4210142575"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5082146003","display_name":"\u041c\u0430\u0442\u0442\u0438 \u0412\u0443\u043e\u0440\u0438\u043d\u0435\u043d","orcid":"https://orcid.org/0000-0002-1734-8228","observed_orcids":["https://orcid.org/0000-0002-1734-8228"]},"institutions":[{"id":"https://openalex.org/I155660961","display_name":"University of Turku","ror":"https://ror.org/05vghhr25","country_code":"FI","type":"education","lineage":["https://openalex.org/I155660961"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Matti Vuorinen","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Turku, Turku, Finland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Turku, Turku, Finland","institution_ids":["https://openalex.org/I155660961"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5089699151"],"corresponding_institution_ids":["https://openalex.org/I155660961"],"apc_list":{"value":3190,"currency":"USD","value_usd":3190},"apc_paid":{"value":3190,"currency":"USD","value_usd":3190},"fwci":2.9768,"has_fulltext":true,"cited_by_count":20,"citation_normalized_percentile":{"value":0.90419162,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":88,"max":98},"biblio":{"volume":"19","issue":"1","first_page":"135","last_page":"155"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9829000234603882,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9829000234603882,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.002899999963119626,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":0.002899999963119626,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/spherical-mirrors","display_name":"spherical mirrors","score":0.8769999742507935},{"id":"https://openalex.org/keywords/light-reflection","display_name":"light reflection","score":0.8450000286102295},{"id":"https://openalex.org/keywords/quartic-equations","display_name":"quartic equations","score":0.9419999718666077},{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"symbolic computation","score":0.843999981880188},{"id":"https://openalex.org/keywords/ancient-optics","display_name":"ancient optics","score":0.574999988079071},{"id":"https://openalex.org/keywords/ray-tracing","display_name":"ray tracing","score":0.39100000262260437},{"id":"https://openalex.org/keywords/catadioptric-optics","display_name":"catadioptric optics","score":0.9449999928474426}],"concepts":[{"id":"https://openalex.org/C153396827","wikidata":"https://www.wikidata.org/wiki/Q1142960","display_name":"Catadioptric system","level":3,"score":0.8531286716461182},{"id":"https://openalex.org/C46984397","wikidata":"https://www.wikidata.org/wiki/Q7257100","display_name":"Ptolemy's table of chords","level":2,"score":0.7404469847679138},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.6273494362831116},{"id":"https://openalex.org/C110277352","wikidata":"https://www.wikidata.org/wiki/Q467980","display_name":"Geometrical optics","level":2,"score":0.5971853733062744},{"id":"https://openalex.org/C121483023","wikidata":"https://www.wikidata.org/wiki/Q7298343","display_name":"Ray tracing (physics)","level":2,"score":0.5766527652740479},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5473246574401855},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5203390121459961},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4901042878627777},{"id":"https://openalex.org/C81253797","wikidata":"https://www.wikidata.org/wiki/Q1506584","display_name":"Curved mirror","level":2,"score":0.48487553000450134},{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.46589720249176025},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3642747700214386},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.3559865355491638},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.310627818107605},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.30410099029541016},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.22948798537254333},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19744446873664856},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1272386908531189},{"id":"https://openalex.org/C15336307","wikidata":"https://www.wikidata.org/wiki/Q1766051","display_name":"Lens (geology)","level":2,"score":0.08339986205101013},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1007/s40315-018-0257-z","doi":"https://doi.org/10.1007/s40315-018-0257-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s40315-018-0257-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s40315-018-0257-z.pdf","source":{"id":"https://openalex.org/S2483979929","display_name":"Computational Methods and Function Theory","issn_l":"1617-9447","issn":["1617-9447","2195-3724"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods and Function Theory","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1706.06924","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1706.06924","pdf_url":"https://arxiv.org/pdf/1706.06924","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:utupub.fi:11111/46973","doi":null,"is_oa":true,"landing_page_url":"https://www.utupub.fi/handle/11111/46973","pdf_url":"https://www.utupub.fi/bitstreams/b02175d4-c43d-4f51-a1eb-01e34b7071e1/download","source":{"id":"https://openalex.org/S4306402470","display_name":"UTUPub (University of Turku)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I155660961","host_organization_name":"University of Turku","host_organization_lineage":["https://openalex.org/I155660961"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://www.utupub.fi/handle/10024/185437","raw_type":null},{"id":"pmh:oai:www.utupub.fi:10024/185437","doi":null,"is_oa":true,"landing_page_url":"https://www.utupub.fi/handle/10024/185437","pdf_url":"https://www.utupub.fi/bitstream/10024/185437/1/Fujimura2019_Article_ThePtolemyAlhazenProblemAndSph.pdf","source":{"id":"https://openalex.org/S4306402470","display_name":"UTUPub (University of Turku)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I155660961","host_organization_name":"University of Turku","host_organization_lineage":["https://openalex.org/I155660961"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},{"id":"doi:10.48550/arxiv.1706.06924","doi":"https://doi.org/10.48550/arxiv.1706.06924","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1706.06924","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.1007/s40315-018-0257-z","doi":"https://doi.org/10.1007/s40315-018-0257-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s40315-018-0257-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s40315-018-0257-z.pdf","source":{"id":"https://openalex.org/S2483979929","display_name":"Computational Methods and Function Theory","issn_l":"1617-9447","issn":["1617-9447","2195-3724"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods and Function Theory","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G8285828049","display_name":"Analysis of dynamics of rational maps using the method of computer algebra","funder_award_id":"15K04943","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320322754","display_name":"Kansainv\u00e4lisen Liikkuvuuden ja Yhteisty\u00f6n Keskus","ror":"https://ror.org/00wj0f678"},{"id":"https://openalex.org/F4320323313","display_name":"Turun Yliopisto","ror":"https://ror.org/05vghhr25"},{"id":"https://openalex.org/F4320331826","display_name":"Turun Yliopistos\u00e4\u00e4ti\u00f6","ror":"https://ror.org/02kjggt26"},{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963351908.pdf","grobid_xml":"https://content.openalex.org/works/W2963351908.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W139827475","https://openalex.org/W1055287694","https://openalex.org/W1522797534","https://openalex.org/W1538409627","https://openalex.org/W1832577878","https://openalex.org/W1970642807","https://openalex.org/W2028268998","https://openalex.org/W2038619192","https://openalex.org/W2078365455","https://openalex.org/W2080393846","https://openalex.org/W2094150771","https://openalex.org/W2142893983","https://openalex.org/W2155348479","https://openalex.org/W2269030894","https://openalex.org/W2530341725","https://openalex.org/W2914661649","https://openalex.org/W2914720498","https://openalex.org/W2994790045","https://openalex.org/W3143622134","https://openalex.org/W3144894925"],"related_works":["https://openalex.org/W3209666400","https://openalex.org/W7140985284","https://openalex.org/W3108777053","https://openalex.org/W1518839473","https://openalex.org/W7025070875","https://openalex.org/W4411975345","https://openalex.org/W2119666631","https://openalex.org/W1967535006","https://openalex.org/W3012158972","https://openalex.org/W7000228089"],"abstract_inverted_index":{"An":[0],"ancient":[1],"optics":[2],"problem":[3,13,67],"of":[4,17,28,55,71],"Ptolemy,":[5],"studied":[6,47],"later":[7],"by":[8],"Alhazen,":[9],"is":[10],"discussed.":[11],"This":[12],"deals":[14],"with":[15,50],"reflection":[16],"light":[18],"in":[19,48,68],"spherical":[20],"mirrors.":[21],"Mathematically,":[22],"this":[23,66],"reduces":[24],"to":[25,65],"the":[26,72],"solution":[27],"a":[29,38],"quartic":[30],"equation,":[31],"which":[32],"we":[33,62],"solve":[34],"and":[35,58],"analyze":[36],"using":[37],"symbolic":[39],"computation":[40],"software.":[41],"Similar":[42],"problems":[43],"have":[44],"been":[45],"recently":[46],"connection":[49],"ray-tracing,":[51],"catadioptric":[52],"optics,":[53],"scattering":[54],"electromagnetic":[56],"waves,":[57],"mathematical":[59],"billiards,":[60],"but":[61],"were":[63],"led":[64],"our":[69],"study":[70],"so-called":[73],"triangular":[74],"ratio":[75],"metric.":[76]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":3},{"year":2018,"cited_by_count":1}],"updated_date":"2026-11-14T11:04:58.165337","created_date":"2019-07-30T00:00:00"}