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[{"key": "dc.contributor.advisor", "value": "Luisto, Rami", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kunttu, Veronika", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-14T13:05:42Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-14T13:05:42Z", "language": null, "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/123456789/64650", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on n\u00e4ytt\u00e4\u00e4, kuinka kolmiulotteisen reaaliavaruuden rotaatiot voidaan esitt\u00e4\u00e4 kvaternioita k\u00e4ytt\u00e4en. Kvaterniot miellet\u00e4\u00e4n kompleksilukujen joukon laajennukseksi, mist\u00e4 syyst\u00e4 tutkielman aluksi tarkastellaan kompleksilukujen joukkoa ja kompleksilukujen yhteytt\u00e4 tason kiertoihin. Kun yhteys kompleksilukujen ja tason kiertojen v\u00e4lill\u00e4 on osoitettu, on luontevaa siirty\u00e4 tarkastelemaan vastaavaa yhteytt\u00e4 kvaternioiden ja avaruuden rotaatioiden v\u00e4lill\u00e4. Kvaternioiden joukossa on kompleksilukujen joukosta tutun imaginaariyksik\u00f6n i lis\u00e4ksi kaksi muuta imaginaarista yksikk\u00f6\u00e4, j ja k. Kvaterniot ovat muotoa q = a1 + bi + cj + dk, miss\u00e4 luvut a, b, c ja d ovat reaalilukuja. Edelleen kvaternioiden osajoukkoja ovat puhtaiden kvaternioiden joukko Im(\u210d) = {bi + cj + dk : b, c, d \u2208 \u211d} sek\u00e4 reaalikvaternioiden joukko Re(\u210d) = {a1: a \u2208 \u211d}. Erityisen kiinnostuneita t\u00e4ss\u00e4 tutkielmassa ollaan puhtaiden kvaternioiden konjugointikuvauksesta, sill\u00e4 puhtaiden kvaternioiden konjugointi vastaa kolmiulotteisen reaaliavaruuden rotaatioita. Kvaternioita k\u00e4ytet\u00e4\u00e4n avaruuden rotaatioiden tarkastelussa, koska monimutkaisten kiertojen tarkastelu helpottuu huomattavasti kvaternioiden konjugointikuvauksen avulla. Avaruuden kierron m\u00e4\u00e4ritt\u00e4miseksi konjugointikuvauksen avulla riitt\u00e4\u00e4, ett\u00e4 tiedet\u00e4\u00e4n rotaation kiertoakseli u sek\u00e4 kiertokulma \u03b8, sill\u00e4 kiertokulman ja -akselin avulla saadaan selville kvaterniokonjugoinnin m\u00e4\u00e4r\u00e4\u00e4v\u00e4 kvaternio t.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-06-14T13:05:42Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-14T13:05:42Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "57", "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.subject.other", "value": "kvaterniot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "avaruuden rotaatiot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Kvaterniot ja niiden yhteys avaruuden rotaatioihin", "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-201906143241", "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": "kompleksiluvut", "language": null, "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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