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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hyv\u00f6nen, Jussi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-12-24T14:57:16Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-12-24T14:57:16Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1809834", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/56542", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sittell\u00e4\u00e4n projektiivista geometriaa aksioomien ja mallien kautta. Tutkielma keskittyy p\u00e4\u00e4asiassa \u00e4\u00e4rellisiin projektiivisiin geometrioihin ja niist\u00e4 erityisesti tasogeometriaan. Tutkielmassa luodaan kirjallisuuskatsaus projektiivisen geometrian alkuvaiheiden kautta aksioomaj\u00e4rjestelm\u00e4n luomiseen ja p\u00e4\u00e4tyen tutustumaan yksinkertaisimpiin malleihin projektiiviselta tasolta. Samalla tulee todistettua my\u00f6s projektiivisen geometrian peruslause ja tutustuttua projektiivisen geometrian hy\u00f6dyllisyyteen k\u00e4sitelt\u00e4ess\u00e4 euklidisen geometrian tilanteita. Euklidisessa geometriassa vallitsevia lauseita voidaan nimitt\u00e4in tulkita projektiivisessa geometriassa ja t\u00e4ll\u00f6in projektiivisen geometrian ominaisuudet mahdollistavat todistusten huomattavan yksinkertaistamisen. Tutkielman viimeisessa kappaleessa on esimerkkein\u00e4 t\u00e4st\u00e4 k\u00e4sittelyss\u00e4 Desarguesin ja Pappusin lauseet. \n\nProjektiivisen geometrian tutkimuksen voidaan katsoa alkaneen jo 1400-luvulla kuvataiteessa ilmenneiden ongelmien seurauksena. Kuvataiteilijat halusivat l\u00f6yt\u00e4\u00e4 yh\u00e4 parempia keinoja taltioida maailmaa mahdollisimman realistisen n\u00e4k\u00f6isen\u00e4 maalauskankaalle ja t\u00e4ss\u00e4 huomattiin perspektiivisest\u00e4 tarkastelusta olevan huomattavaa hy\u00f6ty\u00e4. Ala on sen j\u00e4lkeen kehittynyt syk\u00e4yksitt\u00e4in, kunnes lopulta on voitu todeta kyseess\u00e4 olevan t\u00e4ysin oma geometriansa toimivine aksioomaj\u00e4rjestelmineen. \n\nProjektiivisen geometrian suurin ero euklidiseen geometriaan on paralleeliaksiooman puuttuminen. N\u00e4in ollen projektiivisessa geometriassa mitk\u00e4 tahansa kaksi suoraa leikkaavat toisensa jossain pisteess\u00e4. Euklidisen geometrian yhdensuuntaisia suoria vastaavien suorien leikkauspiste sijaitsee \u00e4\u00e4rett\u00f6myydess\u00e4 ja siit\u00e4 k\u00e4ytet\u00e4\u00e4n nimityst\u00e4 ideaalipiste. Ideaalipisteit\u00e4 ja ideaalipisteiden muodostamaa suoraa voidaan tutkia algebrallisesti yhdess\u00e4 muiden pisteiden ja suorien kanssa k\u00e4ytt\u00e4m\u00e4ll\u00e4 hyv\u00e4ksi homogeenisi\u00e4 koordinaatteja, joihin tutustutaan my\u00f6s t\u00e4ss\u00e4 tutkielmassa. \n\nT\u00e4m\u00e4n tutkielman keski\u00f6ss\u00e4 ovat Rey Cassen ja David Brannanin teokset projektiivisesta geometriasta. Tutkielmassa esitetty projektiivisen geometrian aksioomaj\u00e4rjestelm\u00e4 on Rey Cassen muotoilema. My\u00f6s hieman eri tavalla muotoiltuja, mutta yht\u00e4pit\u00e4vi\u00e4 versioita on julkaistu.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Jussi Hyv\u00f6nen (jutahyvo) on 2017-12-24 14:57:15.172553. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2017-12-24T14:57:16Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201712244881.pdf: 740037 bytes, checksum: a2b781c2e1be9968ae52c98cd9f941eb (MD5)\nlicense.html: 4783 bytes, checksum: 1028613fe33616484e6a24c4e4e6945a (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-12-24T14:57:16Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201712244881.pdf: 740037 bytes, checksum: a2b781c2e1be9968ae52c98cd9f941eb (MD5)\nlicense.html: 4783 bytes, checksum: 1028613fe33616484e6a24c4e4e6945a (MD5)\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (32 sivua)", "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": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "projektiivinen geometria", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "tasogeometria", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": 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