{"id":"https://openalex.org/W21028567","doi":null,"title":"Base points, resultants, and the implicit representation of rational surfaces","display_name":"Base points, resultants, and the implicit representation of rational surfaces","publication_year":1990,"publication_date":"1990-01-01","ids":{"openalex":"https://openalex.org/W21028567","mag":"21028567"},"language":"en","primary_location":{"id":"mag:21028567","doi":null,"is_oa":false,"landing_page_url":"http://dl.acm.org/citation.cfm?id=916992","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},"volume":null,"type":"article","indexed_in":[],"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/A5061689148","display_name":"Eng-Wee Chionh","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Eng-Wee Chionh","raw_affiliation_strings":["Univ. of Waterloo,Waterloo,Ont.,Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univ. of Waterloo,Waterloo,Ont.,Canada","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5061689148"],"corresponding_institution_ids":["https://openalex.org/I151746483"],"apc_list":null,"apc_paid":null,"fwci":6.837,"has_fulltext":false,"cited_by_count":41,"citation_normalized_percentile":{"value":0.97464988,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9839000105857849,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9839000105857849,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.013500000350177288,"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/T10481","display_name":"Computer Graphics and Visualization Techniques","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/rational-surfaces","display_name":"rational surfaces","score":0.8999999761581421},{"id":"https://openalex.org/keywords/implicitization","display_name":"implicitization","score":0.8109999895095825},{"id":"https://openalex.org/keywords/base-points","display_name":"base points","score":0.871999979019165},{"id":"https://openalex.org/keywords/dixon-resultant","display_name":"Dixon resultant","score":0.9509999752044678},{"id":"https://openalex.org/keywords/computer-aided-geometric-design","display_name":"computer-aided geometric design","score":0.6520000100135803},{"id":"https://openalex.org/keywords/solid-modeling","display_name":"solid modeling","score":0.8259999752044678}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6873365640640259},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.6741585731506348},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5933310985565186},{"id":"https://openalex.org/C2777979628","wikidata":"https://www.wikidata.org/wiki/Q208756","display_name":"Rational surface","level":3,"score":0.5719506144523621},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.545049250125885},{"id":"https://openalex.org/C188032258","wikidata":"https://www.wikidata.org/wiki/Q1128371","display_name":"Constructive solid geometry","level":2,"score":0.5363186001777649},{"id":"https://openalex.org/C202887219","wikidata":"https://www.wikidata.org/wiki/Q3895221","display_name":"Parametrization (atmospheric modeling)","level":3,"score":0.5362799167633057},{"id":"https://openalex.org/C2187777","wikidata":"https://www.wikidata.org/wiki/Q1071743","display_name":"Parametric surface","level":3,"score":0.4851006269454956},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4642859697341919},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.42603591084480286},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4105144143104553},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.30631160736083984},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23301419615745544},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.15448608994483948},{"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/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C82706917","wikidata":"https://www.wikidata.org/wiki/Q10251","display_name":"Plasma","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","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},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:21028567","doi":null,"is_oa":false,"landing_page_url":"http://dl.acm.org/citation.cfm?id=916992","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2043206651","https://openalex.org/W1501327712","https://openalex.org/W2045615359","https://openalex.org/W2069497227","https://openalex.org/W1989043587","https://openalex.org/W12091192","https://openalex.org/W2093473068","https://openalex.org/W2884646533","https://openalex.org/W2087794615","https://openalex.org/W2953055098"],"abstract_inverted_index":{"The":[0],"implicit":[1,39,91,106,196,232,255],"representation":[2,92,107],"of":[3,58,71,78,85,93,104,122,128,159,188,194,202,246,271,274],"rational":[4,95,189,248],"surfaces":[5],"is":[6,219],"investigated.":[7],"Associated":[8],"with":[9,152],"a":[10,94,221,251],"parametrization":[11,218,223],"are":[12,30,44,62,102,109,243,268],"two":[13,239],"fundamental":[14,186],"concepts:":[15],"the":[16,25,34,38,63,69,90,105,120,139,155,192,195,200,210,217,225,231,244,247,272,280],"auxiliary":[17,26,35,211,226],"resultant":[18,36,65,212,227],"and":[19,68,137,157,224,250,256,262,282],"base":[20,32,45,72,79,203,237,275],"points.":[21,204,276],"Base":[22],"points":[23,80],"annihilate":[24],"resultant.":[27],"When":[28],"there":[29,43],"no":[31],"points,":[33,46],"gives":[37],"representation;":[40],"but":[41],"when":[42,209],"we":[47],"have":[48],"to":[49,82,111,166],"account":[50],"for":[51,66,88,177,287],"their":[52],"effects":[53],"properly.":[54],"Two":[55],"illuminating":[56],"examples":[57],"this":[59,100,144],"proper":[60],"accounting":[61],"Dixon":[64],"implicitization":[67,265],"definition":[70],"point":[73],"multiplicity.":[74],"A":[75],"careful":[76],"treatment":[77],"leads":[81],"rigorous":[83],"derivations":[84],"constructive":[86],"methods":[87,266],"finding":[89],"surface.":[96],"Also":[97],"following":[98],"from":[99],"analysis":[101],"properties":[103,158,187],"that":[108,208],"pertinent":[110],"CAGD":[112],"as":[113,115,119],"well":[114],"additional":[116],"results":[117,241],"such":[118],"detection":[121],"degenerate":[123],"surface":[124,222,249],"parametrizations.\r\nThis":[125],"thesis":[126,281],"consists":[127],"eight":[129],"chapters.":[130],"Chapter":[131,147,161,181,205,234,259,277],"1":[132],"surveys":[133],"recent":[134],"related":[135],"work":[136],"presents":[138,283],"larger":[140],"context":[141],"in":[142,171,199],"which":[143,267],"research":[145],"belongs.":[146],"2":[148],"provides":[149],"mathematical":[150],"preliminaries,":[151],"emphasis":[153],"on":[154],"construction":[156],"resultants.":[160],"3":[162],"applies":[163],"multivariate":[164],"resultants":[165],"solve":[167],"an":[168],"important":[169,240],"problem":[170],"solid":[172],"modeling--a":[173],"new":[174],"intersection":[175],"algorithm":[176],"three":[178],"quadric":[179],"surfaces.":[180],"4":[182],"establishes":[183],"rigorously":[184],"some":[185,284],"surfaces,":[190],"including":[191],"existence":[193],"equation":[197],"even":[198],"presence":[201,273],"5":[206],"shows":[207],"does":[213],"not":[214],"vanish":[215],"identically,":[216],"indeed":[220],"reveals":[228],"much":[229],"about":[230],"representation.":[233],"6":[235],"discusses":[236],"points;":[238],"here":[242],"incompleteness":[245],"precise":[252],"relationship":[253],"between":[254],"parametric":[257],"degrees.":[258],"7":[260],"describes":[261],"compares":[263],"six":[264],"applicable":[269],"regardless":[270],"8":[278],"concludes":[279],"open":[285],"problems":[286],"future":[288],"research.":[289]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1}],"updated_date":"2026-08-08T07:41:36.138363","created_date":"2016-06-24T00:00:00"}