{"id":"https://openalex.org/W2132148545","doi":"https://doi.org/10.1007/s10444-009-9138-3","title":"Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves","display_name":"Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves","publication_year":2009,"publication_date":"2009-08-06","ids":{"openalex":"https://openalex.org/W2132148545","doi":"https://doi.org/10.1007/s10444-009-9138-3","mag":"2132148545"},"language":"en","primary_location":{"id":"doi:10.1007/s10444-009-9138-3","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-009-9138-3","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9138-3.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Computational Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9138-3.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113866987","display_name":"Rida T. Farouki","orcid":null},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Rida T. Farouki","raw_affiliation_strings":["Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, USA","Department of Mechanical and Aeronautical Engineering, University of California, Davis, USA 95616#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, USA","institution_ids":["https://openalex.org/I84218800"]},{"raw_affiliation_string":"Department of Mechanical and Aeronautical Engineering, University of California, Davis, USA 95616#TAB#","institution_ids":["https://openalex.org/I84218800"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5113866987"],"corresponding_institution_ids":["https://openalex.org/I84218800"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":4.2622,"has_fulltext":true,"cited_by_count":41,"citation_normalized_percentile":{"value":0.94823543,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"33","issue":"3","first_page":"331","last_page":"348"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9994000196456909,"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":0.9994000196456909,"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/T10571","display_name":"Robotic Mechanisms and Dynamics","score":0.9836999773979187,"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"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9287999868392944,"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/quaternion","display_name":"Quaternion","score":0.8269551992416382},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7861195802688599},{"id":"https://openalex.org/keywords/quintic-function","display_name":"Quintic function","score":0.5755651593208313},{"id":"https://openalex.org/keywords/rotation","display_name":"Rotation (mathematics)","score":0.4808981418609619},{"id":"https://openalex.org/keywords/hopf-fibration","display_name":"Hopf fibration","score":0.4406462609767914},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.42047107219696045},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.38811489939689636},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3869222402572632},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3644605278968811},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.22715777158737183}],"concepts":[{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.8269551992416382},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7861195802688599},{"id":"https://openalex.org/C124966035","wikidata":"https://www.wikidata.org/wiki/Q33104507","display_name":"Quintic function","level":3,"score":0.5755651593208313},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.4808981418609619},{"id":"https://openalex.org/C55232844","wikidata":"https://www.wikidata.org/wiki/Q1627604","display_name":"Hopf fibration","level":2,"score":0.4406462609767914},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.42047107219696045},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38811489939689636},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3869222402572632},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3644605278968811},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.22715777158737183},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s10444-009-9138-3","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-009-9138-3","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9138-3.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:escholarship.org/ark:/13030/qt7qb9973s","is_oa":false,"landing_page_url":"https://escholarship.org/uc/item/7qb9973s","pdf_url":null,"source":{"id":"https://openalex.org/S4306400115","display_name":"eScholarship (California Digital Library)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2801248553","host_organization_name":"California Digital Library","host_organization_lineage":["https://openalex.org/I2801248553"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Advances in Computational Mathematics, vol 33, iss 3","raw_type":"article"},{"id":"pmh:qt7qb9973s","is_oa":false,"landing_page_url":"http://www.escholarship.org/uc/item/7qb9973s","pdf_url":null,"source":{"id":"https://openalex.org/S4306400115","display_name":"eScholarship (California Digital Library)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2801248553","host_organization_name":"California Digital Library","host_organization_lineage":["https://openalex.org/I2801248553"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Farouki, Rida T.(2010). Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves. Advances in Computational Mathematics, 33(3), pp 331-348. doi: 10.1007/s10444-009-9138-3. Retrieved from: http://www.escholarship.org/uc/item/7qb9973s","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1007/s10444-009-9138-3","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-009-9138-3","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-009-9138-3.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11","score":0.8100000023841858}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2132148545.pdf","grobid_xml":"https://content.openalex.org/works/W2132148545.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W56095176","https://openalex.org/W140657133","https://openalex.org/W169274659","https://openalex.org/W1532721371","https://openalex.org/W1570562679","https://openalex.org/W1980328962","https://openalex.org/W1993671632","https://openalex.org/W2009566988","https://openalex.org/W2024715780","https://openalex.org/W2036543490","https://openalex.org/W2042662795","https://openalex.org/W2047301367","https://openalex.org/W2053578036","https://openalex.org/W2053753540","https://openalex.org/W2057978774","https://openalex.org/W2063216546","https://openalex.org/W2072181956","https://openalex.org/W2076155370","https://openalex.org/W2077590674","https://openalex.org/W2082719777","https://openalex.org/W2090107137","https://openalex.org/W2090267299","https://openalex.org/W2150526134","https://openalex.org/W2346131912","https://openalex.org/W2531272619","https://openalex.org/W3118818388","https://openalex.org/W4245211388","https://openalex.org/W4388321161"],"related_works":["https://openalex.org/W4254786939","https://openalex.org/W4238758668","https://openalex.org/W3127350447","https://openalex.org/W3108700312","https://openalex.org/W2887325221","https://openalex.org/W4225472102","https://openalex.org/W2382948573","https://openalex.org/W2465700534","https://openalex.org/W2382267983","https://openalex.org/W2091132606"],"abstract_inverted_index":{"Simple":[0],"criteria":[1,36],"for":[2,26,114,134,145],"the":[3,20,48,75,80,84,98,122,135,141],"existence":[4],"of":[5,18,40,89,104,130,143],"rational":[6],"rotation-minimizing":[7],"frames":[8],"(RRMFs)":[9],"on":[10,74,97,109],"quintic":[11],"space":[12],"curves":[13,29,117],"are":[14,45,52],"determined,":[15],"in":[16,30,47,87,152],"terms":[17,88],"both":[19,33],"quaternion":[21,60,77,123],"and":[22,51,92,118,124,149,160],"Hopf":[23,81,125],"map":[24,82,126],"representations":[25],"Pythagorean-hodograph":[27],"(PH)":[28],"\u211d3.":[31],"In":[32,59,79],"cases,":[34],"these":[35,105,131],"amount":[37],"to":[38],"satisfaction":[39],"three":[41,76],"scalar":[42],"constraints":[43,106],"that":[44],"quadratic":[46,71,95],"curve":[49],"coefficients,":[50],"thus":[53],"much":[54],"simpler":[55],"than":[56],"previous":[57],"criteria.":[58],"form,":[61,83],"RRMF":[62,136],"quintics":[63,137],"can":[64],"be":[65],"characterized":[66],"by":[67],"just":[68],"a":[69,111],"single":[70],"(vector)":[72],"constraint":[73,96],"coefficients.":[78,101],"characterization":[85],"is":[86,107],"one":[90,93],"real":[91],"complex":[94,100],"six":[99],"The":[102,128],"identification":[103],"based":[108],"introducing":[110],"\u201ccanonical":[112],"form\u201d":[113],"spatial":[115,157],"PH":[116],"judicious":[119],"transformations":[120],"between":[121],"descriptions.":[127],"simplicity":[129],"new":[132],"characterizations":[133],"should":[138],"help":[139],"facilitate":[140],"development":[142],"algorithms":[144],"their":[146],"construction,":[147],"analysis,":[148],"practical":[150],"use":[151],"applications":[153],"such":[154],"as":[155],"animation,":[156],"motion":[158],"planning,":[159],"swept":[161],"surface":[162],"constructions.":[163]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}