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[{"key": "dc.contributor.advisor", "value": "\u00c4kkinen, Tuomo", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Rantakyl\u00e4, Heini", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-05-20T06:04:07Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-05-20T06:04:07Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/102463", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Fermat-Torricelli ongelmassa tulee l\u00f6yt\u00e4\u00e4 sellainen piste, jossa et\u00e4isyydet annetusta pisteest\u00e4 kolmion k\u00e4rkiin yhteenlaskettuna tulee olla mahdollisimman pieni. T\u00e4m\u00e4 ongelma on esiintynyt alun perin 1600-luvulla matemaatikko Fermat'n kirjoituksissa. Ajan saatossa ongelmalle on syntynyt erilaisia nimityksi\u00e4 ja sovelluksia.\nT\u00e4ss\u00e4 lopputy\u00f6ss\u00e4 Fermat-Torricelli ongelma ratkaistaan ensin tasossa kahdella eri tavalla, joista ensimm\u00e4inen ratkaisu mukailee Torricellin konstruktiota ja toinen ratkaisuista niin kutsuttua unkarilaista todistusta. Molemmissa todistuksissa merkitt\u00e4v\u00e4ss\u00e4 asemassa on alkuper\u00e4isen kolmion ymp\u00e4rille piirretty apukolmio. Todistuksissa p\u00e4\u00e4dyt\u00e4\u00e4n ratkaisuun, miss\u00e4 minimointipisteest\u00e4 n\u00e4hd\u00e4\u00e4n annetun kolmion sivut 120 asteen kulmassa.\nTasossa ongelma laajennetaan painotettuun malliin, miss\u00e4 et\u00e4isyydet kolmion k\u00e4rjest\u00e4 toivottuun minimipisteeseen saavat kaikki omat kertoimensa. Pisteen sijainti todistetaan ensin geometrisesti konstruoiden. Konstruktiossa luodaan apukolmio, jonka sivut riippuvat et\u00e4isyyksien kertoimista. Ratkaisuksi saadaan piste, josta n\u00e4hd\u00e4\u00e4n alkuper\u00e4isen kolmion sivut sellaisessa kulmassa, joka on riippuvainen apukolmion kulmien suuruudesta. Lis\u00e4ksi ongelmalle annetaan mekaaninen ratkaisu, jossa et\u00e4isyyksien kertoimet ovat kuvattuina p\u00f6yt\u00e4tason l\u00e4pi menevien painojen suuruuksina. Mekaanisessa ratkaisussakin p\u00e4\u00e4dyt\u00e4\u00e4n pisteeseen, joka riippuu et\u00e4isyyksien kertoimista.\nTy\u00f6n viimeisess\u00e4 osassa Fermat-Torricelli ongelma ratkaistaan ensin \u00e4\u00e4rellisess\u00e4 avaruudessa kolmella pisteell\u00e4 ja lopuksi \u00e4\u00e4rellisell\u00e4 m\u00e4\u00e4r\u00e4ll\u00e4 pisteit\u00e4. Ensimm\u00e4isess\u00e4 tapauksessa pisteen sijainti ratkaistaan konveksi analyysin keinoin hy\u00f6dynt\u00e4en gradienttia. Toisessa tapauksessa pisteen sijainnin m\u00e4\u00e4rittelemiseen hy\u00f6dynnet\u00e4\u00e4n Weiszfeldin algoritmia. Algoritmi osoitetaan toimivaksi suppenevien osajonojen avulla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-05-20T06:04:07Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-05-20T06:04:07Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "41", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Fermat-Torricelli", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Fermat-Torricelli ongelma", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202505204397", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "unknown accessibility", "language": "en", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "ei tietoa saavutettavuudesta", "language": "fi", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}]
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