{"id":"https://openalex.org/W7223255166","doi":"https://doi.org/10.48550/arxiv.2610.28689","title":"Approximating the Ellipse Perimeter by \"Bending the Line\": The ELIP Closed-Form Family","display_name":"Approximating the Ellipse Perimeter by \"Bending the Line\": The ELIP Closed-Form Family","publication_year":2026,"publication_date":"2026-10-20","ids":{"openalex":"https://openalex.org/W7223255166","doi":"https://doi.org/10.48550/arxiv.2610.28689"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.28689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.28689","pdf_url":"https://arxiv.org/pdf/2610.28689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.28689","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159273020","display_name":"Salvador Eugenio Ayala-Raggi","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I721619","display_name":"Benem\u00e9rita Universidad Aut\u00f3noma de Puebla","ror":"https://ror.org/03p2z7827","country_code":"MX","type":"education","lineage":["https://openalex.org/I721619"]}],"countries":["MX"],"is_corresponding":false,"raw_author_name":"Ayala-Raggi, Salvador Eugenio","raw_affiliation_strings":["Facultad de Ciencias de la Electr\u00f3nica , Benem\u00e9rita Universidad Aut\u00f3noma de Puebla (BUAP) , Puebla , M\u00e9xico"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Facultad de Ciencias de la Electr\u00f3nica , Benem\u00e9rita Universidad Aut\u00f3noma de Puebla (BUAP) , Puebla , M\u00e9xico","institution_ids":["https://openalex.org/I721619"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5132969808","display_name":"Manuel Rend\u00f3n-Mar\u00edn","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I721619","display_name":"Benem\u00e9rita Universidad Aut\u00f3noma de Puebla","ror":"https://ror.org/03p2z7827","country_code":"MX","type":"education","lineage":["https://openalex.org/I721619"]}],"countries":["MX"],"is_corresponding":false,"raw_author_name":"Rend\u00f3n-Mar\u00edn, Manuel","raw_affiliation_strings":["Facultad de Ciencias de la Electr\u00f3nica , Benem\u00e9rita Universidad Aut\u00f3noma de Puebla (BUAP) , Puebla , M\u00e9xico"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Facultad de Ciencias de la Electr\u00f3nica , Benem\u00e9rita Universidad Aut\u00f3noma de Puebla (BUAP) , Puebla , M\u00e9xico","institution_ids":["https://openalex.org/I721619"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I721619"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9218999743461609,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9218999743461609,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T13806","display_name":"Mathematical and Computational Methods","score":0.04320000112056732,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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":0.013500000350177288,"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/ellipse-perimeter","display_name":"ellipse perimeter","score":0.984000027179718},{"id":"https://openalex.org/keywords/closed-form-approximation","display_name":"closed-form approximation","score":0.7979999780654907},{"id":"https://openalex.org/keywords/linear-interpolation","display_name":"linear interpolation","score":0.5569999814033508},{"id":"https://openalex.org/keywords/logarithmic-asymptotics","display_name":"logarithmic asymptotics","score":0.8479999899864197},{"id":"https://openalex.org/keywords/error-bounds","display_name":"error bounds","score":0.3880000114440918}],"concepts":[{"id":"https://openalex.org/C74261601","wikidata":"https://www.wikidata.org/wiki/Q40112","display_name":"Ellipse","level":2,"score":0.8118000030517578},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7860000133514404},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5993000268936157},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5382000207901001},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.5235999822616577},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5116000175476074},{"id":"https://openalex.org/C98503990","wikidata":"https://www.wikidata.org/wiki/Q28474","display_name":"Perimeter","level":2,"score":0.41659998893737793},{"id":"https://openalex.org/C186080144","wikidata":"https://www.wikidata.org/wiki/Q358733","display_name":"Series expansion","level":2,"score":0.40610000491142273},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38659998774528503},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.3547999858856201},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.3440999984741211},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.34279999136924744},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.33239999413490295},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.33219999074935913},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.3142000138759613},{"id":"https://openalex.org/C2776716506","wikidata":"https://www.wikidata.org/wiki/Q5161158","display_name":"Conjugate points","level":2,"score":0.31349998712539673},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.31189998984336853},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.30070000886917114},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.29420000314712524},{"id":"https://openalex.org/C205890394","wikidata":"https://www.wikidata.org/wiki/Q48297","display_name":"Parabola","level":2,"score":0.29260000586509705},{"id":"https://openalex.org/C11683690","wikidata":"https://www.wikidata.org/wiki/Q752726","display_name":"Asymptotic expansion","level":2,"score":0.29170000553131104},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.29170000553131104},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2847999930381775},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.28360000252723694},{"id":"https://openalex.org/C195597517","wikidata":"https://www.wikidata.org/wiki/Q438842","display_name":"Alternating series","level":3,"score":0.2791999876499176},{"id":"https://openalex.org/C108598597","wikidata":"https://www.wikidata.org/wiki/Q124255","display_name":"Conic section","level":2,"score":0.2646999955177307},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.25360000133514404}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.28689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.28689","pdf_url":"https://arxiv.org/pdf/2610.28689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.28689","doi":"https://doi.org/10.48550/arxiv.2610.28689","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.28689","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.28689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.28689","pdf_url":"https://arxiv.org/pdf/2610.28689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223255166.pdf","grobid_xml":"https://content.openalex.org/works/W7223255166.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W4403544954","https://openalex.org/W7150722928","https://openalex.org/W7155984905","https://openalex.org/W7172539201"],"related_works":["https://openalex.org/W7214723219","https://openalex.org/W6931034022","https://openalex.org/W4410573982","https://openalex.org/W7223339917","https://openalex.org/W4226263852","https://openalex.org/W7149686485","https://openalex.org/W7166812381","https://openalex.org/W4416245565","https://openalex.org/W4403544954","https://openalex.org/W2979496966"],"abstract_inverted_index":{"The":[0,67,168],"perimeter":[1,157],"of":[2,13,63,90,109],"an":[3],"ellipse":[4],"is":[5],"expressed":[6],"exactly":[7],"through":[8,159],"the":[9,14,33,37,59,143,155,162],"complete":[10],"elliptic":[11],"integral":[12],"second":[15],"kind,":[16],"motivating":[17],"accurate":[18],"closed-form":[19],"approximations.":[20],"We":[21],"introduce":[22],"ELIP":[23,99,121],"(Ellipse":[24],"Linear":[25],"Interpolated":[26],"Perimeter),":[27],"a":[28,112,171],"family":[29],"built":[30],"by":[31,73,80,131],"correcting":[32],"straight":[34],"line":[35],"joining":[36],"two":[38],"exact":[39,60,132,156,175],"normalized":[40],"endpoint":[41,176],"values.":[42],"Two":[43],"branches":[44],"are":[45,129],"developed:":[46],"logarithmic-power":[47],"corrections":[48,57],"with":[49,105],"optional":[50],"deformation":[51],"and":[52,55,95,184],"exponential":[53],"modulators,":[54],"rational-logarithmic":[56],"incorporating":[58],"asymptotic":[61],"behavior":[62],"highly":[64],"elongated":[65],"ellipses.":[66],"framework":[68],"distinguishes":[69],"coefficients":[70,128],"fixed":[71],"analytically":[72],"Maclaurin":[74,133],"matching":[75],"from":[76],"parameters":[77],"optimized":[78],"numerically":[79],"minimax":[81],"fitting.":[82],"Compact":[83],"rationalized":[84],"formulas":[85],"achieve":[86],"maximum":[87,114],"relative":[88,115],"errors":[89],"approximately":[91],"6.84,":[92],"3.57,":[93],"0.904,":[94],"1.596":[96],"ppm.":[97,119,151],"An":[98],"R5,2":[100],"+":[101,123],"3":[102,124],"log":[103],"formula":[104,153],"three":[106],"fitted":[107],"degrees":[108],"freedom":[110],"has":[111],"certified":[113,144],"error":[116,146],"below":[117,149],"0.001623":[118],"For":[120],"R5,3":[122],"log,":[125],"all":[126],"rational-core":[127],"determined":[130],"matching.":[134],"A":[135],"one-parameter":[136],"multiplicative":[137],"correction,":[138],"1":[139],"-":[140],"A*h^12*(1-h)^(7/4),":[141],"reduces":[142],"global":[145],"bound":[147],"to":[148],"0.000452":[150],"This":[152],"matches":[154],"series":[158,179],"h^8,":[160],"while":[161],"correction":[163],"starts":[164],"at":[165],"order":[166],"h^12.":[167],"results":[169],"provide":[170],"systematic":[172],"approach":[173],"combining":[174],"geometry,":[177],"local":[178],"constraints,":[180],"elongated-limit":[181],"logarithmic":[182],"asymptotics,":[183],"low-dimensional":[185],"numerical":[186],"optimization.":[187]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-23T00:00:00"}