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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pynssi, Maija", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-27T10:09:39Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-27T10:09:39Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/64872", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielmassa tutustutaan Steinerin sis\u00e4ellipsiin. Steinerin sis\u00e4ellipsiksi kutsutaan\r\nkolmion sis\u00e4ll\u00e4 olevaa ellipsi\u00e4, joka sivuaa kolmion jokaista sivua sivun keskipisteess\u00e4.\r\nSteinerin sis\u00e4ellipsi on ympyr\u00e4 jos ja vain jos kolmio on tasasivuinen.\r\n\r\nTutkielman p\u00e4\u00e4tulokset ovat Steinerin lause ja Mardenin lause. Steinerin lauseen\r\nmukaan jokaisella kolmiolla on yksik\u00e4sitteinen Steinerin sis\u00e4ellipsi. Mardenin lauseessa saadaan Steinerin sis\u00e4ellipsin polttopisteet kompleksitason kolmannen asteen polynomin, jonka juuria ovat kolmion k\u00e4rkipisteet, kriittisist\u00e4 pisteist\u00e4. T\u00e4st\u00e4 my\u00f6s\r\nhuomataan, ett\u00e4 Steinerin sis\u00e4ellipsi on ympyr\u00e4 jos ja vain jos polynomin derivaatalla on kaksoisjuuri. T\u00e4ll\u00f6in kolmio on tasasivuinen. Lis\u00e4ksi tutkielmassa osoitetaan,\r\nett\u00e4 Steinerin sis\u00e4ellipsi on pinta-alaltaan suurin mahdollinen ellipsi, joka voidaan\r\nkonstruoida kolmion sis\u00e4lle.\r\n\r\nTutkielmassa my\u00f6s tutustutaan Steinerin yksikk\u00f6sis\u00e4ellipsiin, joka on Steinerin sis\u00e4ellipsin erikoistilanne. T\u00e4ll\u00f6in kolmion k\u00e4rkipisteet ovat yksikk\u00f6ympyr\u00e4ll\u00e4. Lopuksi\r\nk\u00e4yd\u00e4\u00e4n l\u00e4pi yll\u00e4tt\u00e4vi\u00e4kin tuloksia, kun huomataan geometriasta tuttujen Fermat\u2019n\r\npisteiden yhteys Steinerin sis\u00e4ellipsien akseleihin. Lis\u00e4ksi Fermat\u2019n pisteiden avulla\r\npystyt\u00e4\u00e4n konstruoimaan Steinerin sis\u00e4ellipsin polttopisteet.\r\n\r\nSteinerin ja Mardenin lauseiden todistamista varten k\u00e4ytet\u00e4\u00e4n kompleksia\ufb03ineja kuvauksia. Kompleksia\ufb03init kuvaukset kuvaavat kolmiot kolmioiksi ja s\u00e4ilytt\u00e4v\u00e4t\r\njanojen keskipisteet. Kompleksia\ufb03ineilla kuvauksilla Steinerin sis\u00e4ellipsi kuvautuu\r\nkuvautuneen kolmion Steinerin sis\u00e4ellipsiksi. N\u00e4in ollen kompleksia\ufb03ineilla kuvauksilla pystyt\u00e4\u00e4n siirt\u00e4m\u00e4\u00e4n, skaalaamaan ja kiert\u00e4m\u00e4\u00e4n kolmiota haluttuun paikkaan,\r\njolloin Mardenin lauseen todistaminen helpottuu alkuper\u00e4isen polynomin yksinkertaistuessa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2019-06-27T10:09:39Z\r\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-27T10:09:39Z (GMT). No. of bitstreams: 0\r\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Steinerin sis\u00e4ellipsi", "language": "", "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-201906273475", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "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": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "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": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "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.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "polynomit", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "polttopisteet", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kompleksiluvut", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ellipsit (geometriset kuviot)", "language": "", "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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