{"id":"https://openalex.org/W7162509882","doi":"https://doi.org/10.4230/lipics.socg.2026.84","title":"The Voronoi Diagram of Four Lines in \u211d\u00b3","display_name":"The Voronoi Diagram of Four Lines in \u211d\u00b3","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7162509882","doi":"https://doi.org/10.4230/lipics.socg.2026.84"},"language":"en","primary_location":{"id":"pmh:oai:drops-oai.dagstuhl.de:25891","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.84","pdf_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol367-socg2026/LIPIcs.SoCG.2026.84/LIPIcs.SoCG.2026.84.pdf","source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol367-socg2026/LIPIcs.SoCG.2026.84/LIPIcs.SoCG.2026.84.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046472103","display_name":"Evanthia Papadopoulou","orcid":"https://orcid.org/0000-0003-0144-7384"},"institutions":[{"id":"https://openalex.org/I57201433","display_name":"Universit\u00e0 della Svizzera italiana","ror":"https://ror.org/03c4atk17","country_code":"CH","type":"education","lineage":["https://openalex.org/I57201433"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Papadopoulou, Evanthia","raw_affiliation_strings":["Faculty of Informatics, Universit\u00e0 della Svizzera italiana (USI), Lugano, Switzerland"],"raw_orcid":"https://orcid.org/0000-0003-0144-7384","affiliations":[{"raw_affiliation_string":"Faculty of Informatics, Universit\u00e0 della Svizzera italiana (USI), Lugano, Switzerland","institution_ids":["https://openalex.org/I57201433"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5115596280","display_name":"Zeyu Wang","orcid":"https://orcid.org/0009-0005-0595-6663"},"institutions":[{"id":"https://openalex.org/I57201433","display_name":"Universit\u00e0 della Svizzera italiana","ror":"https://ror.org/03c4atk17","country_code":"CH","type":"education","lineage":["https://openalex.org/I57201433"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Wang, Zeyu","raw_affiliation_strings":["Faculty of Informatics, Universit\u00e0 della Svizzera italiana (USI), Lugano, Switzerland"],"raw_orcid":"https://orcid.org/0009-0004-4207-198X","affiliations":[{"raw_affiliation_string":"Faculty of Informatics, Universit\u00e0 della Svizzera italiana (USI), Lugano, Switzerland","institution_ids":["https://openalex.org/I57201433"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.85856304,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.503000020980835,"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"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.503000020980835,"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"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.1356000006198883,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.06419999897480011,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/voronoi-diagram","display_name":"Voronoi diagram","score":0.9649999737739563},{"id":"https://openalex.org/keywords/power-diagram","display_name":"Power diagram","score":0.7190999984741211},{"id":"https://openalex.org/keywords/weighted-voronoi-diagram","display_name":"Weighted Voronoi diagram","score":0.6050000190734863},{"id":"https://openalex.org/keywords/diagram","display_name":"Diagram","score":0.5422000288963318},{"id":"https://openalex.org/keywords/centroidal-voronoi-tessellation","display_name":"Centroidal Voronoi tessellation","score":0.5149999856948853},{"id":"https://openalex.org/keywords/position","display_name":"Position (finance)","score":0.5012999773025513},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4934999942779541},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.4794999957084656},{"id":"https://openalex.org/keywords/twist","display_name":"Twist","score":0.4311000108718872}],"concepts":[{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.9649999737739563},{"id":"https://openalex.org/C20573064","wikidata":"https://www.wikidata.org/wiki/Q13410436","display_name":"Power diagram","level":3,"score":0.7190999984741211},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6485999822616577},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6053000092506409},{"id":"https://openalex.org/C20468658","wikidata":"https://www.wikidata.org/wiki/Q17130624","display_name":"Weighted Voronoi diagram","level":3,"score":0.6050000190734863},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.5422000288963318},{"id":"https://openalex.org/C205672865","wikidata":"https://www.wikidata.org/wiki/Q5062961","display_name":"Centroidal Voronoi tessellation","level":3,"score":0.5149999856948853},{"id":"https://openalex.org/C198082294","wikidata":"https://www.wikidata.org/wiki/Q3399648","display_name":"Position (finance)","level":2,"score":0.5012999773025513},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4934999942779541},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4794999957084656},{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.4311000108718872},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.4104999899864197},{"id":"https://openalex.org/C182124507","wikidata":"https://www.wikidata.org/wiki/Q166154","display_name":"Line segment","level":2,"score":0.3921999931335449},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.374099999666214},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3725999891757965},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30720001459121704},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.29910001158714294},{"id":"https://openalex.org/C185877587","wikidata":"https://www.wikidata.org/wiki/Q1757407","display_name":"Medial axis","level":2,"score":0.28619998693466187},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2825999855995178},{"id":"https://openalex.org/C2779304628","wikidata":"https://www.wikidata.org/wiki/Q3503480","display_name":"Face (sociological concept)","level":2,"score":0.2721000015735626},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C199845137","wikidata":"https://www.wikidata.org/wiki/Q145490","display_name":"Network topology","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2535000145435333},{"id":"https://openalex.org/C113238511","wikidata":"https://www.wikidata.org/wiki/Q1071612","display_name":"k-nearest neighbors algorithm","level":2,"score":0.2529999911785126}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:drops-oai.dagstuhl.de:25891","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.84","pdf_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol367-socg2026/LIPIcs.SoCG.2026.84/LIPIcs.SoCG.2026.84.pdf","source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},{"id":"doi:10.4230/lipics.socg.2026.84","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.socg.2026.84","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":""}],"best_oa_location":{"id":"pmh:oai:drops-oai.dagstuhl.de:25891","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.84","pdf_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol367-socg2026/LIPIcs.SoCG.2026.84/LIPIcs.SoCG.2026.84.pdf","source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7162509882.pdf","grobid_xml":"https://content.openalex.org/works/W7162509882.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,31,63,222],"consider":[1],"the":[2,10,22,35,40,69,113,116,122,125,142,149,162,217,224],"Voronoi":[3,41,127,183,240],"diagram":[4,42,70,184],"of":[5,18,25,37,43,79,88,115,129,152,161,185,198,211,238],"lines":[6,27,45,187],"in":[7,21,28,39,46,139,166,173,188,191,215],"\u211d\u00b3":[8,192],"under":[9],"Euclidean":[11],"metric,":[12],"and":[13,54,56,94,124,148,219,235],"give":[14],"a":[15,65,73,77,101,167,208],"full":[16,93,98,120,203],"classification":[17,225],"its":[19],"structure":[20,67],"base":[23],"case":[24],"four":[26,44,130,186],"general":[29,47,189],"position.":[30],"first":[32],"show":[33],"that":[34],"number":[36,151],"vertices":[38,153,214],"position":[48,190],"is":[49,76,100,154],"always":[50],"even,":[51],"between":[52],"0":[53],"8,":[55],"all":[57,146,180],"such":[58],"numbers":[59],"can":[60,106,193],"be":[61,107,194],"realized.":[62],"identify":[64],"key":[66],"for":[68],"formation,":[71],"called":[72],"twist,":[74],"which":[75,105,137],"pair":[78],"consecutive":[80],"intersections":[81],"among":[82],"trisector":[83],"branches;":[84],"only":[85],"two":[86,213],"types":[87],"twists":[89],"are":[90,138,145,171,179],"possible,":[91],"so-called":[92],"partial":[95],"twists.":[96],"A":[97],"twist":[99,206],"purely":[102],"local":[103],"structure,":[104],"inserted":[108],"or":[109],"removed":[110],"without":[111],"affecting":[112],"rest":[114],"diagram.":[117],"Assuming":[118],"no":[119],"twists,":[121],"nearest":[123,218],"farthest":[126,163,220],"diagrams":[128],"lines,":[131],"each":[132,205],"have":[133],"15":[134],"distinct":[135],"topologies,":[136],"one-to-one":[140,174],"correspondence;":[141],"two-dimensional":[143,168],"faces":[144],"unbounded,":[147],"total":[150],"at":[155],"most":[156],"six.":[157],"The":[158,176],"unbounded":[159],"features":[160],"diagram,":[164],"encoded":[165],"spherical":[169],"map,":[170],"also":[172],"correspondence.":[175],"identified":[177],"topologies":[178,200],"realizable.":[181],"Any":[182],"obtained":[195],"from":[196],"one":[197],"these":[199],"by":[201,226],"inserting":[202],"twists;":[204],"induces":[207],"bounded":[209],"face":[210],"exactly":[212],"both":[216],"diagrams.":[221,241],"obtain":[223],"an":[227],"exhaustive":[228],"search":[229],"algorithm":[230],"using":[231],"some":[232],"new":[233],"structural":[234],"combinatorial":[236],"observations":[237],"line":[239]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-28T00:00:00"}
