Shortest paths in arbitrary plane domains

Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2019-03
Main Authors: Hoehn, L C, Oversteegen, L G, Tymchatyn, E D
Format: Article
Language:English
Subjects:
Domains
Shortest path planning
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites
container_end_page
container_issue
container_start_page
container_title arXiv.org
container_volume
creator Hoehn, L C
Oversteegen, L G
Tymchatyn, E D
description Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with endpoints fixed whose intermediate paths \(h_t\), for \(t \in [0,1)\), satisfy \(h_t((0,1)) \subset \Omega\). We prove this result even in the case when there is no path of finite Euclidean length homotopic to \(\gamma\) under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.
doi_str_mv 10.48550/arxiv.1903.06737
format article
fullrecord <record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2459481563</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2459481563</sourcerecordid><originalsourceid>FETCH-LOGICAL-a523-d1e11bb3ccf5cd52fe5eac35fbd84320e8837ceea06fd43a7c895898d6967bbc3</originalsourceid><addsrcrecordid>eNotjstKAzEUQIMgWGo_wF3AlYsZk9zcJLOU4gsKLuy-3LzolDozJlPRv7egq7M75zB2I0WrHaK4p_Ldf7WyE9AKY8FesIUCkI3TSl2xVa0HIYQyViHCgt2978cypzrzieZ95f3Aqfh-LlR--HSkIfE4flA_1Gt2melY0-qfS7Z9etyuX5rN2_Pr-mHTECpookxSeg8hZAwRVU6YKABmH50GJZJzYENKJEyOGsgG16HrXDSdsd4HWLLbP-1Uxs_TeWx3GE9lOBd3SmOnnUQD8AteOEN5</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2459481563</pqid></control><display><type>article</type><title>Shortest paths in arbitrary plane domains</title><source>Publicly Available Content Database</source><creator>Hoehn, L C ; Oversteegen, L G ; Tymchatyn, E D</creator><creatorcontrib>Hoehn, L C ; Oversteegen, L G ; Tymchatyn, E D</creatorcontrib><description>Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with endpoints fixed whose intermediate paths \(h_t\), for \(t \in [0,1)\), satisfy \(h_t((0,1)) \subset \Omega\). We prove this result even in the case when there is no path of finite Euclidean length homotopic to \(\gamma\) under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.1903.06737</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Domains ; Shortest path planning</subject><ispartof>arXiv.org, 2019-03</ispartof><rights>2019. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.proquest.com/docview/2459481563?pq-origsite=primo$$EHTML$$P50$$Gproquest$$Hfree_for_read</linktohtml><link.rule.ids>780,784,25753,27925,37012,44590</link.rule.ids></links><search><creatorcontrib>Hoehn, L C</creatorcontrib><creatorcontrib>Oversteegen, L G</creatorcontrib><creatorcontrib>Tymchatyn, E D</creatorcontrib><title>Shortest paths in arbitrary plane domains</title><title>arXiv.org</title><description>Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with endpoints fixed whose intermediate paths \(h_t\), for \(t \in [0,1)\), satisfy \(h_t((0,1)) \subset \Omega\). We prove this result even in the case when there is no path of finite Euclidean length homotopic to \(\gamma\) under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.</description><subject>Domains</subject><subject>Shortest path planning</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>PIMPY</sourceid><recordid>eNotjstKAzEUQIMgWGo_wF3AlYsZk9zcJLOU4gsKLuy-3LzolDozJlPRv7egq7M75zB2I0WrHaK4p_Ldf7WyE9AKY8FesIUCkI3TSl2xVa0HIYQyViHCgt2978cypzrzieZ95f3Aqfh-LlR--HSkIfE4flA_1Gt2melY0-qfS7Z9etyuX5rN2_Pr-mHTECpookxSeg8hZAwRVU6YKABmH50GJZJzYENKJEyOGsgG16HrXDSdsd4HWLLbP-1Uxs_TeWx3GE9lOBd3SmOnnUQD8AteOEN5</recordid><startdate>20190315</startdate><enddate>20190315</enddate><creator>Hoehn, L C</creator><creator>Oversteegen, L G</creator><creator>Tymchatyn, E D</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20190315</creationdate><title>Shortest paths in arbitrary plane domains</title><author>Hoehn, L C ; Oversteegen, L G ; Tymchatyn, E D</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a523-d1e11bb3ccf5cd52fe5eac35fbd84320e8837ceea06fd43a7c895898d6967bbc3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Domains</topic><topic>Shortest path planning</topic><toplevel>online_resources</toplevel><creatorcontrib>Hoehn, L C</creatorcontrib><creatorcontrib>Oversteegen, L G</creatorcontrib><creatorcontrib>Tymchatyn, E D</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>ProQuest Central Essentials</collection><collection>AUTh Library subscriptions: ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>SciTech Premium Collection (Proquest) (PQ_SDU_P3)</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hoehn, L C</au><au>Oversteegen, L G</au><au>Tymchatyn, E D</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Shortest paths in arbitrary plane domains</atitle><jtitle>arXiv.org</jtitle><date>2019-03-15</date><risdate>2019</risdate><eissn>2331-8422</eissn><abstract>Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with endpoints fixed whose intermediate paths \(h_t\), for \(t \in [0,1)\), satisfy \(h_t((0,1)) \subset \Omega\). We prove this result even in the case when there is no path of finite Euclidean length homotopic to \(\gamma\) under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.1903.06737</doi><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier EISSN: 2331-8422
ispartof arXiv.org, 2019-03
issn 2331-8422
language eng
recordid cdi_proquest_journals_2459481563
source Publicly Available Content Database
subjects Domains
Shortest path planning
title Shortest paths in arbitrary plane domains
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-07T14%3A28%3A37IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Shortest%20paths%20in%20arbitrary%20plane%20domains&rft.jtitle=arXiv.org&rft.au=Hoehn,%20L%20C&rft.date=2019-03-15&rft.eissn=2331-8422&rft_id=info:doi/10.48550/arxiv.1903.06737&rft_dat=%3Cproquest%3E2459481563%3C/proquest%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-a523-d1e11bb3ccf5cd52fe5eac35fbd84320e8837ceea06fd43a7c895898d6967bbc3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2459481563&rft_id=info:pmid/&rfr_iscdi=true