{"id":"https://openalex.org/W2946298966","doi":"https://doi.org/10.2478/forma-2019-0005","title":"Cross-Ratio in Real Vector Space","display_name":"Cross-Ratio in Real Vector Space","publication_year":2019,"publication_date":"2019-04-01","ids":{"openalex":"https://openalex.org/W2946298966","doi":"https://doi.org/10.2478/forma-2019-0005","mag":"2946298966"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2019-0005","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0005","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p47.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p47.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068514239","display_name":"Roland Coghetto","orcid":"https://orcid.org/0000-0002-4901-0766"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Roland Coghetto","raw_affiliation_strings":["Rue de la Brasserie 5, 7100 La Louvi\u00e8re , Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Rue de la Brasserie 5, 7100 La Louvi\u00e8re , Belgium","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5068514239"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.1448037,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"27","issue":"1","first_page":"47","last_page":"60"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9872000217437744,"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/T12504","display_name":"Mathematics and Applications","score":0.9824000000953674,"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/mathematics","display_name":"Mathematics","score":0.5952584147453308},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5703051686286926},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5404634475708008},{"id":"https://openalex.org/keywords/real-line","display_name":"Real line","score":0.4644218981266022},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.43151822686195374},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.42055612802505493},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.41259950399398804},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.3431527614593506},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.22032326459884644},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.14007386565208435}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5952584147453308},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5703051686286926},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5404634475708008},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.4644218981266022},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.43151822686195374},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.42055612802505493},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.41259950399398804},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.3431527614593506},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.22032326459884644},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.14007386565208435},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2019-0005","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0005","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p47.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:8f5e88a1ba084fca88f2870c9bdb96bf","is_oa":false,"landing_page_url":"https://doaj.org/article/8f5e88a1ba084fca88f2870c9bdb96bf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 27, Iss 1, Pp 47-60 (2019)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/7839","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/7839","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2019-0005","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0005","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p47.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2946298966.pdf","grobid_xml":"https://content.openalex.org/works/W2946298966.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W365379550","https://openalex.org/W1751036247","https://openalex.org/W1934555024","https://openalex.org/W2138253682","https://openalex.org/W2159772409","https://openalex.org/W2295728669","https://openalex.org/W2502358513","https://openalex.org/W2741396296","https://openalex.org/W2887099240","https://openalex.org/W4239806157"],"related_works":["https://openalex.org/W2980963558","https://openalex.org/W1882246857","https://openalex.org/W3103304230","https://openalex.org/W2185360082","https://openalex.org/W1992365905","https://openalex.org/W4287664878","https://openalex.org/W3212023493","https://openalex.org/W1991411408","https://openalex.org/W2066456778","https://openalex.org/W4288278856"],"abstract_inverted_index":{"Summary":[0],"Using":[1],"Mizar":[2,242,310],"[1],":[3],"in":[4,70,256,304],"the":[5,14,30,38,41,48,51,56,61,80,85,88,92,106,241,245,257,261,272,290,305,309,317,322,325],"context":[6],"of":[7,16,19,32,40,43,50,64,66,99,102,308,316,324],"a":[8,71,100,286,332],"real":[9,72,81,86,107,263],"vector":[10,73,82,246,264,274],"space,":[11],"we":[12,59,76],"introduce":[13,60],"concept":[15],"affine":[17],"ratio":[18],"three":[20],"aligned":[21,69],"points":[22,68,104],"(see":[23],"[5]).":[24],"It":[25],"is":[26,84,122],"also":[27],"equivalent":[28],"to":[29,37,47,91,288],"notion":[31,42,63,89,94],"\u201cMesure":[33],"alg\u00e8brique\u201d":[34],"1":[35,112],",":[36,113,116,119],"opposite":[39,49],"Teilverh\u00e4ltnis":[44],"2":[45,115],"or":[46],"ordered":[52],"length-ratio":[53],"[9].":[54],"In":[55,240],"second":[57],"part,":[58],"classic":[62],"\u201ccross-ratio\u201d":[65],"4":[67,121,296],"space.":[74],"Finally,":[75],"show":[77],"that":[78],"if":[79],"space":[83,265,275],"line,":[87],"corresponds":[90],"classical":[93],"3":[95,118],"[9]:":[96],"The":[97,298,314],"cross-ratio":[98,318],"quadruple":[101],"distinct":[103],"on":[105],"line":[108],"with":[109],"coordinates":[110],"x":[111,114,117,120],"given":[123],"by:":[124],"<m:math":[125],"xmlns:m=\"http://www.w3.org/1998/Math/MathML\">":[126],"<m:mrow>":[127,130,137,144,151,159,161,168,174,176,183,192,194,201,207,209,216],"<m:mo>(</m:mo>":[128],"<m:msub>":[129,136,143,150,160,167,175,182,193,200,208,215],"<m:mi>x</m:mi>":[131,138,145,152,162,169,177,184,195,202,210,217],"</m:mrow>":[132,139,146,153,163,170,173,178,185,188,196,203,206,211,218,221,223],"<m:mn>1</m:mn>":[133,171,219],"</m:msub>":[134,141,148,155,165,172,180,187,198,205,213,220],"<m:mo>,</m:mo>":[135,149],"<m:mn>2</m:mn>":[140,186,204],"<m:mo>;</m:mo>":[142],"<m:mn>3</m:mn>":[147,164,179],"<m:mn>4</m:mn>":[154,197,212],"<m:mo>)</m:mo>":[156],"<m:mo>=</m:mo>":[157],"<m:mfrac>":[158,191],"<m:mo>-</m:mo>":[166,181,199,214],"</m:mfrac>":[189,222],"<m:mo>.</m:mo>":[190],"</m:math>":[224],"$$({x_1},{x_2};{x_3},{x_4})":[225],"=":[226],"{{{x_3}":[227],"-":[228,232,234,238],"{x_1}}":[229],"\\over":[230,236],"{{x_3}":[231],"{x_2}}}.{{{x_4}":[233],"{x_2}}":[235],"{{x_4}":[237],"{x_1}}}$$":[239],"Mathematical":[243],"Library,":[244],"spaces":[247],"were":[248],"first":[249],"defined":[250,267,277],"by":[251,268,278,293],"Kusak,":[252],"Leonczuk":[253],"and":[254,271,282],"Muzalewski":[255],"article":[258],"[6],":[259],"while":[260],"actual":[262],"was":[266,276],"Trybulec":[269],"[10]":[270],"complex":[273],"Endou":[279],"[4].":[280],"Nakasho":[281],"Shidama":[283],"have":[284],"developed":[285],"solution":[287],"explore":[289],"notions":[291],"introduced":[292],"different":[294],"authors":[295],"[7].":[297],"definitions":[299],"can":[300],"be":[301],"directly":[302],"linked":[303],"HTMLized":[306],"version":[307],"library":[311],"5":[312],".":[313],"study":[315],"will":[319],"continue":[320],"within":[321],"framework":[323],"Klein-":[326],"Beltrami":[327],"model":[328],"[2],":[329],"[3].":[330],"For":[331],"generalized":[333],"cross-ratio,":[334],"see":[335],"Papadopoulos":[336],"[8].":[337]},"counts_by_year":[],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2019-05-29T00:00:00"}