{"id":"https://openalex.org/W7213346776","doi":"https://doi.org/10.48550/arxiv.2609.14653","title":"Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane","display_name":"Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane","publication_year":2026,"publication_date":"2026-09-13","ids":{"openalex":"https://openalex.org/W7213346776","doi":"https://doi.org/10.48550/arxiv.2609.14653"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.14653","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14653","pdf_url":"https://arxiv.org/pdf/2609.14653","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/2609.14653","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036202949","display_name":"Micha\u0142 Zwierzy\u0144ski","orcid":"https://orcid.org/0000-0002-9627-1563","observed_orcids":["https://orcid.org/0000-0002-9627-1563"]},"institutions":[{"id":"https://openalex.org/I108403487","display_name":"Warsaw University of Technology","ror":"https://ror.org/00y0xnp53","country_code":"PL","type":"education","lineage":["https://openalex.org/I108403487"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Zwierzy\u0144ski, Micha\u0142","raw_affiliation_strings":["Warsaw University of Technology Faculty of Mathematics and Information Science ul . Koszykowa 75 00-662 Warsaw , Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Warsaw University of Technology Faculty of Mathematics and Information Science ul . Koszykowa 75 00-662 Warsaw , Poland","institution_ids":["https://openalex.org/I108403487"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5036202949"],"corresponding_institution_ids":["https://openalex.org/I108403487"],"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.6220999956130981,"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"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.6220999956130981,"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"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.1160999983549118,"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"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0494999997317791,"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/frontal","display_name":"frontal","score":0.6880000233650208},{"id":"https://openalex.org/keywords/cusps","display_name":"cusps","score":0.6600000262260437},{"id":"https://openalex.org/keywords/parallel-curves","display_name":"parallel curves","score":0.527999997138977},{"id":"https://openalex.org/keywords/singularities","display_name":"singularities","score":0.5260000228881836},{"id":"https://openalex.org/keywords/ovals","display_name":"ovals","score":0.39800000190734863}],"concepts":[{"id":"https://openalex.org/C2779567845","wikidata":"https://www.wikidata.org/wiki/Q3492725","display_name":"Caustic (mathematics)","level":2,"score":0.7749000191688538},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7513999938964844},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.6459000110626221},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.5812000036239624},{"id":"https://openalex.org/C535169671","wikidata":"https://www.wikidata.org/wiki/Q17080006","display_name":"Semiclassical physics","level":3,"score":0.5254999995231628},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.48739999532699585},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4569999873638153},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4361000061035156},{"id":"https://openalex.org/C133978748","wikidata":"https://www.wikidata.org/wiki/Q15955882","display_name":"Reflection symmetry","level":2,"score":0.40290001034736633},{"id":"https://openalex.org/C58315980","wikidata":"https://www.wikidata.org/wiki/Q179436","display_name":"Asymptote","level":2,"score":0.38830000162124634},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.38530001044273376},{"id":"https://openalex.org/C200917960","wikidata":"https://www.wikidata.org/wiki/Q3847057","display_name":"Singular point of a curve","level":2,"score":0.3506999909877777},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33869999647140503},{"id":"https://openalex.org/C96403706","wikidata":"https://www.wikidata.org/wiki/Q543956","display_name":"Monodromy","level":2,"score":0.3366999924182892},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.33469998836517334},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.3248000144958496},{"id":"https://openalex.org/C43809302","wikidata":"https://www.wikidata.org/wiki/Q1060874","display_name":"Plane curve","level":2,"score":0.323199987411499},{"id":"https://openalex.org/C44306375","wikidata":"https://www.wikidata.org/wiki/Q902019","display_name":"Symmetry group","level":2,"score":0.3181000053882599},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.30570000410079956},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.28130000829696655},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C139210041","wikidata":"https://www.wikidata.org/wiki/Q2145840","display_name":"Resonance (particle physics)","level":2,"score":0.2799000144004822},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.27950000762939453},{"id":"https://openalex.org/C16345878","wikidata":"https://www.wikidata.org/wiki/Q107472979","display_name":"Orientation (vector space)","level":2,"score":0.27459999918937683},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.2623000144958496},{"id":"https://openalex.org/C40425006","wikidata":"https://www.wikidata.org/wiki/Q7201020","display_name":"Plane symmetry","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.2596000134944916},{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.25699999928474426},{"id":"https://openalex.org/C20892832","wikidata":"https://www.wikidata.org/wiki/Q3271879","display_name":"Circular symmetry","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C188374626","wikidata":"https://www.wikidata.org/wiki/Q5914418","display_name":"Mirror symmetry","level":2,"score":0.25220000743865967}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.14653","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14653","pdf_url":"https://arxiv.org/pdf/2609.14653","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.2609.14653","doi":"https://doi.org/10.48550/arxiv.2609.14653","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.14653","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.14653","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14653","pdf_url":"https://arxiv.org/pdf/2609.14653","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213346776.pdf","grobid_xml":"https://content.openalex.org/works/W7213346776.grobid-xml"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W990022286","https://openalex.org/W1973259578","https://openalex.org/W1975924433","https://openalex.org/W2008436127","https://openalex.org/W2009108135","https://openalex.org/W2011380711","https://openalex.org/W2018540236","https://openalex.org/W2054747728","https://openalex.org/W2062264317","https://openalex.org/W2090432663","https://openalex.org/W2096153943","https://openalex.org/W2120740205","https://openalex.org/W2135257082","https://openalex.org/W2148652575","https://openalex.org/W2218289401","https://openalex.org/W2282727827","https://openalex.org/W2399476258","https://openalex.org/W2506580371","https://openalex.org/W2519628436","https://openalex.org/W2792153344","https://openalex.org/W2897603741","https://openalex.org/W2948417207","https://openalex.org/W3003638096","https://openalex.org/W3133268527","https://openalex.org/W3211793835","https://openalex.org/W3214501069","https://openalex.org/W4206311976","https://openalex.org/W4287733489","https://openalex.org/W4382061164","https://openalex.org/W4389379510","https://openalex.org/W4391824958","https://openalex.org/W4391988157","https://openalex.org/W4403453674","https://openalex.org/W4406815708","https://openalex.org/W4413825767","https://openalex.org/W4415845029","https://openalex.org/W7115190374","https://openalex.org/W7131298145","https://openalex.org/W7133508418","https://openalex.org/W7134124610","https://openalex.org/W7165632757","https://openalex.org/W7202317656"],"related_works":["https://openalex.org/W4415845029","https://openalex.org/W2897603741","https://openalex.org/W2403097085","https://openalex.org/W7015322794","https://openalex.org/W3211793835","https://openalex.org/W759419585","https://openalex.org/W2012790627","https://openalex.org/W7016102517","https://openalex.org/W2021121248","https://openalex.org/W2009108135"],"abstract_inverted_index":{"Motivated":[0],"by":[1],"the":[2,5,17,34,38,45,48,58,64,87,95,131,138,146,167],"role":[3],"of":[4,37,44,47,52,57,67,94,166],"Wigner":[6,39,132],"caustic":[7,40],"in":[8],"semiclassical":[9],"phase-space":[10],"analysis,":[11],"we":[12],"extend":[13],"it,":[14],"together":[15],"with":[16],"centre":[18,60,139],"symmetry":[19,61,140],"set,":[20],"from":[21],"regular":[22,31],"planar":[23],"curves":[24],"to":[25,142],"cooriented":[26],"frontals.":[27],"For":[28],"an":[29,83],"angularly":[30],"parallel":[32,106],"pair,":[33],"signed":[35,50,169],"speed":[36],"is":[41,113],"one":[42],"half":[43],"difference":[46],"extended":[49],"radii":[51],"curvature,":[53],"while":[54],"singular":[55,110],"points":[56,66],"finite":[59],"set":[62,141],"are":[63],"critical":[65],"their":[68],"projective":[69,156],"ratio.":[70],"These":[71],"formulas":[72],"yield":[73],"criteria":[74],"and":[75,80,90,153,162],"explicit":[76],"invariants":[77],"for":[78,99,150],"ordinary":[79],"higher":[81],"cusps,":[82],"order-lowering":[84],"relation":[85],"between":[86],"two":[88,168],"constructions,":[89],"a":[91,109,115,121,125,128,151,155],"multiplicity-weighted":[92],"extension":[93],"classical":[96],"cusp-count":[97],"inequality":[98],"strictly":[100],"convex":[101],"ovals.":[102],"We":[103,144],"also":[104],"analyse":[105],"pairs":[107],"containing":[108],"frontal":[111],"which":[112,158],"not":[114],"front.":[116],"A":[117],"$5/2$-cusp":[118],"produces":[119],"either":[120],"$5/2$-":[122],"or,":[123],"at":[124],"signed-radius":[126],"resonance,":[127],"$5/3$-cusp":[129,152],"on":[130],"caustic.":[133],"The":[134],"opposite":[135],"resonance":[136],"sends":[137],"infinity.":[143],"obtain":[145],"corresponding":[147],"transfer":[148],"results":[149],"give":[154],"completion":[157],"resolves":[159],"vanishing":[160],"denominators":[161],"simultaneous":[163],"finite-order":[164],"zeros":[165],"radii.":[170]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-16T00:00:00"}
