{"id":"https://openalex.org/W2093946627","doi":"https://doi.org/10.1016/s0167-8396(02)00129-2","title":"Using duality to implicitize and find cusps and inflection points of B\u00e9zier curves","display_name":"Using duality to implicitize and find cusps and inflection points of B\u00e9zier curves","publication_year":2002,"publication_date":"2002-06-01","ids":{"openalex":"https://openalex.org/W2093946627","doi":"https://doi.org/10.1016/s0167-8396(02)00129-2","mag":"2093946627"},"language":"en","primary_location":{"id":"doi:10.1016/s0167-8396(02)00129-2","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0167-8396(02)00129-2","pdf_url":null,"source":{"id":"https://openalex.org/S135221619","display_name":"Computer Aided Geometric Design","issn_l":"0167-8396","issn":["0167-8396","1879-2332"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computer Aided Geometric Design","raw_type":"journal-article"},"type":"article","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/A5040357294","display_name":"Richard R. Patterson","orcid":null},"institutions":[{"id":"https://openalex.org/I55769427","display_name":"Indiana University \u2013 Purdue University Indianapolis","ror":"https://ror.org/05gxnyn08","country_code":"US","type":"education","lineage":["https://openalex.org/I55769427","https://openalex.org/I592451"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Richard R. Patterson","raw_affiliation_strings":["Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202-3216, USA","Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, IN#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202-3216, USA","institution_ids":["https://openalex.org/I55769427"]},{"raw_affiliation_string":"Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, IN#TAB#","institution_ids":["https://openalex.org/I55769427"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5040357294"],"corresponding_institution_ids":["https://openalex.org/I55769427"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.17336256,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"19","issue":"6","first_page":"433","last_page":"444"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9984999895095825,"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9354000091552734,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8599758148193359},{"id":"https://openalex.org/keywords/inflection-point","display_name":"Inflection point","score":0.8538399338722229},{"id":"https://openalex.org/keywords/parametrization","display_name":"Parametrization (atmospheric modeling)","score":0.7404537796974182},{"id":"https://openalex.org/keywords/algebraic-curve","display_name":"Algebraic curve","score":0.6308150887489319},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.543838620185852},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.5143902897834778},{"id":"https://openalex.org/keywords/stable-curve","display_name":"Stable curve","score":0.47399771213531494},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.4134218990802765},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.41154545545578003},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.36834779381752014},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.33697211742401123},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3277449905872345},{"id":"https://openalex.org/keywords/algebraic-surface","display_name":"Algebraic surface","score":0.1164611279964447},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05962646007537842}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8599758148193359},{"id":"https://openalex.org/C74545648","wikidata":"https://www.wikidata.org/wiki/Q212794","display_name":"Inflection point","level":2,"score":0.8538399338722229},{"id":"https://openalex.org/C202887219","wikidata":"https://www.wikidata.org/wiki/Q3895221","display_name":"Parametrization (atmospheric modeling)","level":3,"score":0.7404537796974182},{"id":"https://openalex.org/C207043602","wikidata":"https://www.wikidata.org/wiki/Q266237","display_name":"Algebraic curve","level":2,"score":0.6308150887489319},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.543838620185852},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.5143902897834778},{"id":"https://openalex.org/C184363387","wikidata":"https://www.wikidata.org/wiki/Q3001219","display_name":"Stable curve","level":4,"score":0.47399771213531494},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.4134218990802765},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.41154545545578003},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.36834779381752014},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.33697211742401123},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3277449905872345},{"id":"https://openalex.org/C69653121","wikidata":"https://www.wikidata.org/wiki/Q1434805","display_name":"Algebraic surface","level":3,"score":0.1164611279964447},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05962646007537842},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","level":1,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0},{"id":"https://openalex.org/C74902906","wikidata":"https://www.wikidata.org/wiki/Q1190858","display_name":"Radiative transfer","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/s0167-8396(02)00129-2","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0167-8396(02)00129-2","pdf_url":null,"source":{"id":"https://openalex.org/S135221619","display_name":"Computer Aided Geometric Design","issn_l":"0167-8396","issn":["0167-8396","1879-2332"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computer Aided Geometric Design","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W163123765","https://openalex.org/W575696198","https://openalex.org/W1561016548","https://openalex.org/W1970663370","https://openalex.org/W1971343670","https://openalex.org/W2015941968","https://openalex.org/W2022637219","https://openalex.org/W2029438801","https://openalex.org/W2042171391","https://openalex.org/W2053314097","https://openalex.org/W2064044318","https://openalex.org/W2066706130","https://openalex.org/W2108521527","https://openalex.org/W2301618816","https://openalex.org/W2319466955","https://openalex.org/W2798442789","https://openalex.org/W3110494848","https://openalex.org/W3143314279","https://openalex.org/W3174974282","https://openalex.org/W4234426551","https://openalex.org/W4252725080","https://openalex.org/W6676326272","https://openalex.org/W6697766018","https://openalex.org/W7028097335"],"related_works":["https://openalex.org/W116910719","https://openalex.org/W2443880266","https://openalex.org/W2387929688","https://openalex.org/W3148155780","https://openalex.org/W2952227937","https://openalex.org/W4302340261","https://openalex.org/W2184570656","https://openalex.org/W2913856894","https://openalex.org/W3197207566","https://openalex.org/W2093946627"],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}