{"id":"https://openalex.org/W1987574683","doi":"https://doi.org/10.1145/1390768.1390782","title":"Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities","display_name":"Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities","publication_year":2008,"publication_date":"2008-07-20","ids":{"openalex":"https://openalex.org/W1987574683","doi":"https://doi.org/10.1145/1390768.1390782","mag":"1987574683"},"language":"en","primary_location":{"id":"doi:10.1145/1390768.1390782","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1390768.1390782","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-first international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"},"type":"preprint","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107405594","display_name":"Jean\u2010Charles Faug\u00e8re","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Jean-Charles Faug\u00e8re","raw_affiliation_strings":["INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111488193","display_name":"Guillaume Moroz","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Guillaume Moroz","raw_affiliation_strings":["INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016765679","display_name":"Fabrice Rouillier","orcid":"https://orcid.org/0009-0004-2972-822X"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Fabrice Rouillier","raw_affiliation_strings":["INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048254079","display_name":"Mohab Safey El Din","orcid":"https://orcid.org/0000-0001-9463-1257"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Mohab Safey El Din","raw_affiliation_strings":["INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France","institution_ids":["https://openalex.org/I1326498283"]},{"raw_affiliation_string":"INRIA Paris-Rocquencourt Center SALSA Project, Paris, France#TAB#","institution_ids":["https://openalex.org/I1326498283"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.2102,"has_fulltext":false,"cited_by_count":48,"citation_normalized_percentile":{"value":0.94052056,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"79","last_page":"86"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9991999864578247,"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/T10571","display_name":"Robotic Mechanisms and Dynamics","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discriminant","display_name":"Discriminant","score":0.7396603226661682},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6831613183021545},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6430357694625854},{"id":"https://openalex.org/keywords/gr\u00f6bner-basis","display_name":"Gr\u00f6bner basis","score":0.4432520866394043},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.4306989014148712},{"id":"https://openalex.org/keywords/variety","display_name":"Variety (cybernetics)","score":0.42787665128707886},{"id":"https://openalex.org/keywords/algebraic-variety","display_name":"Algebraic variety","score":0.42348363995552063},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.41273680329322815},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.37011557817459106},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3640296459197998},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.331007182598114},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24293243885040283},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.24008217453956604},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.18212231993675232},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.14285492897033691},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13774707913398743}],"concepts":[{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.7396603226661682},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6831613183021545},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6430357694625854},{"id":"https://openalex.org/C198082693","wikidata":"https://www.wikidata.org/wiki/Q1551631","display_name":"Gr\u00f6bner basis","level":3,"score":0.4432520866394043},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4306989014148712},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.42787665128707886},{"id":"https://openalex.org/C165761256","wikidata":"https://www.wikidata.org/wiki/Q648995","display_name":"Algebraic variety","level":3,"score":0.42348363995552063},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.41273680329322815},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.37011557817459106},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3640296459197998},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.331007182598114},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24293243885040283},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.24008217453956604},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.18212231993675232},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.14285492897033691},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13774707913398743},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1145/1390768.1390782","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1390768.1390782","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-first international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.163.2097","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.2097","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-spiral.lip6.fr/~safey/Articles/p3p_issac08.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.188.678","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.188.678","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-salsa.lip6.fr/%7Ejcf/Papers/ISSAC08.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-01111711v1","is_oa":false,"landing_page_url":"https://inria.hal.science/hal-01111711","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSAC 2008 - 21st International Symposium on Symbolic and Algebraic Computation, Jul 2008, Hagenberg, Austria. pp.79-86, ⟨10.1145/1390768.1390782⟩","raw_type":"Conference papers"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.7300000190734863,"display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W143088124","https://openalex.org/W1520462245","https://openalex.org/W1535871240","https://openalex.org/W1543638401","https://openalex.org/W1570787829","https://openalex.org/W1574151794","https://openalex.org/W1604670202","https://openalex.org/W1977415556","https://openalex.org/W1985565156","https://openalex.org/W1989367821","https://openalex.org/W1991930050","https://openalex.org/W2007438578","https://openalex.org/W2011693299","https://openalex.org/W2012079420","https://openalex.org/W2038345254","https://openalex.org/W2042769738","https://openalex.org/W2082387287","https://openalex.org/W2085261163","https://openalex.org/W2089113283","https://openalex.org/W2097649661","https://openalex.org/W2104223485","https://openalex.org/W2122928074","https://openalex.org/W2128687423","https://openalex.org/W2129613645","https://openalex.org/W2134237713","https://openalex.org/W2141170819","https://openalex.org/W2156916166","https://openalex.org/W2157536469","https://openalex.org/W2788179600","https://openalex.org/W2909934176","https://openalex.org/W4213199372"],"related_works":["https://openalex.org/W3162124064","https://openalex.org/W2953857700","https://openalex.org/W3091355467","https://openalex.org/W4241330371","https://openalex.org/W4389363760","https://openalex.org/W3203470546","https://openalex.org/W2055878129","https://openalex.org/W4299738432","https://openalex.org/W2244854043","https://openalex.org/W4386732654"],"abstract_inverted_index":{"Classifying":[0],"the":[1,8,13,24,31,34,47,69,72,75,79,85,90,99,144,153,157,178,204,237,242,253,275],"Perspective-Three-Point":[2],"problem":[3,86],"(abbreviated":[4],"by":[5,160,240],"P3P":[6],"in":[7,11,121,135,143,151,156,185,274],"sequel)":[9],"consists":[10,119,150],"determining":[12,88],"number":[14],"of":[15,18,27,46,56,78,93,109,139,207,220,258],"possible":[16,91,111],"positions":[17],"a":[19,43,53,104,123,173,194,199],"camera":[20],"with":[21,61,96,177],"respect":[22,97],"to":[23,51,98,129,262],"apparent":[25,76],"position":[26,77],"three":[28,35,80],"points.":[29,163],"In":[30,68,193],"case":[32],"where":[33],"points":[36,81,273],"form":[37],"an":[38],"isosceles":[39],"triangle,":[40],"we":[41,197],"give":[42,103],"full":[44,200,256],"classification":[45,265],"P3P.":[48],"This":[49],"leads":[50],"consider":[52],"polynomial":[54,57],"system":[55,159],"equations":[58],"and":[59,102,226,247,250],"inequalities":[60],"4":[62],"parameters":[63,73,154],"which":[64,187,266],"is":[65,267],"generically":[66],"zero-dimensional.":[67],"present":[70],"situation,":[71],"represent":[74],"so":[82],"that":[83],"solving":[84,116],"means":[87],"all":[89],"numbers":[92],"real":[94,141],"solutions":[95],"parameters'":[100],"values":[101],"sample":[105,162],"point":[106,134,229],"for":[107,115,170,231],"each":[108,136],"these":[110,161],"numbers.":[112],"One":[113],"way":[114],"such":[117,172,211],"systems":[118],"first":[120,195],"computing":[122],"discriminant":[124],"variety.":[125],"Then,":[126],"one":[127,133],"has":[128],"compute":[130],"at":[131],"least":[132],"connected":[137],"component":[138],"its":[140],"complementary":[142],"parameter's":[145,276],"space.":[146,277],"The":[147],"last":[148],"step":[149],"specializing":[152],"appearing":[155],"initial":[158],"Many":[164],"computational":[165],"tools":[166],"may":[167],"be":[168],"used":[169],"implementing":[171],"general":[174],"method,":[175],"starting":[176],"well":[179],"known":[180],"Cylindrical":[181],"Algebraic":[182],"Decomposition":[183],"(CAD":[184],"short),":[186],"provides":[188],"more":[189,270],"information":[190],"than":[191,271],"required.":[192],"stage,":[196],"propose":[198],"algorithm":[201,239],"based":[202],"on":[203],"straightforward":[205],"use":[206],"some":[208],"sophisticated":[209],"software":[210],"as":[212],"FGb":[213],"(Grobner":[214],"bases":[215],"computations)":[216],"RS":[217],"(real":[218],"roots":[219],"zero-dimensional":[221],"systems),":[222],"DV":[223],"(Discriminant":[224],"varieties)":[225],"RAGlib":[227],"(Critical":[228],"methods":[230],"semi-algebraic":[232],"systems).":[233],"We":[234],"then":[235],"improve":[236],"global":[238],"refining":[241],"required":[243],"computable":[244],"mathematical":[245],"objects":[246],"related":[248],"algorithms":[249],"finally":[251],"provide":[252],"classification.":[254],"Three":[255],"days":[257],"computation":[259],"were":[260],"necessary":[261],"get":[263],"this":[264],"obtained":[268],"from":[269],"40000":[272]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":5}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}