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[{"key": "dc.contributor.author", "value": "Tunis, Pernilla", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2013-02-18T15:02:14Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2013-02-18T15:02:14Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2012", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1253823", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/40956", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Denna pro gradu-avhandling behandlar sf\u00e4risk geometri och kartprojektioner. F\u00f6rsta kapitelet ger en introduktion till den icke-euklidiska geometrin sf\u00e4risk geometri. Avst\u00e5nd och vinklar p\u00e5 en sf\u00e4r definieras och det bevisas bl.a att den kortaste v\u00e4gen mellan tv\u00e5 punkter p\u00e5 en sf\u00e4r alltid \u00e4r en storcirkelb\u00e5ge. \n\nHarriot-Girards sats som s\u00e4ger att man kan r\u00e4kna ut en triangels area med hj\u00e4lp av sf\u00e4rens radie och triangelns vinklar bevisas med hj\u00e4lp av m\u00e5nsk\u00e4ror. Med hj\u00e4lp av denna sats f\u00e5s ocks\u00e5 en regel med vars hj\u00e4lp man kan r\u00e4kna ut vinkelsumman i en triangel. \n\nFyra trigonometriska formler f\u00f6r r\u00e4tvinkliga sf\u00e4riska trianglar bevisas med hj\u00e4lp av trigonometriska regler f\u00f6r trianglar i planet. En av dessa formler kallas f\u00f6r Pythagoras sats f\u00f6r sf\u00e4riska trianglar, eftersom man med denna formel kan r\u00e4kna ut en sidas l\u00e4ngd i en triangel om man vet de andra tv\u00e5 sidornas l\u00e4ngder. Inom trigonometrin i planet \u00e4r sinus- och cosinussatsen centrala. Samma regler som anv\u00e4nds i planet kan inte anv\u00e4ndas f\u00f6r att r\u00e4kna ut vinklar och sidor i sf\u00e4riska trianglar, men i st\u00e4llet bevisas sinus- och cosinussatsen f\u00f6r sf\u00e4riska trianglar. Med hj\u00e4lp av cosinussatsen ber\u00e4knas bl.a. avst\u00e5ndet mellan Jyv\u00e4skyl\u00e4 och New York. \n\nAndra kapitlet behandlar kartprojektioner. Med hj\u00e4lp av kartprojektioner kan jorden avbildas p\u00e5 ett plan. Att avbildningen inte kan ske utan formf\u00f6r\u00e4ndring bevisas. Skillnaderna mellan olika kartprojektioner beskrivs och tre kartprojektioner h\u00e4rleds. En av de \u00e4ldsta kartprojektionerna \u00e4r den azimutala kartprojektionen stereografisk projektion. Denna kartprojektionen h\u00e4rleds geometriskt med hj\u00e4lp av r\u00e4tvinkliga trianglar. Det bevisas ocks\u00e5 med hj\u00e4lp av r\u00e4tvinkliga trianglar att den stereografiska projektionen \u00e4r konform. \n\nMercators projektion \u00e4r en konform och cylindrisk kartprojektion, som inte kan h\u00e4rledas geometriskt. Denna projektion h\u00e4rleds p\u00e5 tv\u00e5 olika s\u00e4tt. F\u00f6rst h\u00e4rleds projektionen med hj\u00e4lp av logaritmfunktionen och den stereografiska projektionen. D\u00e4refter h\u00e4rleds projektionen med hj\u00e4lp av villkoret att projektionen \u00e4r konform. En stor f\u00f6rdel med en karta gjord med hj\u00e4lp av Mercator projektion \u00e4r att varje linje representerar en loxodrom p\u00e5 sf\u00e4ren. En loxodrom \u00e4r en kurva p\u00e5 sf\u00e4ren som sk\u00e4r alla meridianer och paralleller med konstant vinkel. Om man hela tiden f\u00f6ljer samma kurs f\u00e4rdas man s\u00e5ledes efter en loxodrom, allts\u00e5 efter en rak linje p\u00e5 en karta gjord med hj\u00e4lp av Mercators projektion. P\u00e5 grund av denna egenskap anv\u00e4nds Mercators projektion ofta n\u00e4r man ritar sj\u00f6kort. P\u00e5 en stereografisk projektion utg\u00f6rs loxodromen av en logaritmisk spiral. Detta bevisas med hj\u00e4lp av exponentialfunktionen. I exemplena ber\u00e4knas bl.a. avst\u00e5nd l\u00e4ngs loxodromen och avst\u00e5nd l\u00e4ngs storcirkelb\u00e5gen. \n\nDen sista kartprojektionen som behandlas \u00e4r Albers projektion. Albers projektion skiljer sig mycket fr\u00e5n Mercators projektion, eftersom Albers projektion inte \u00e4r konform. Albers projektion \u00e4r i st\u00e4llet ytriktig och konisk. Med hj\u00e4lp av dessa villkor h\u00e4rleds projektionen.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Pernilla Tunis (pesokrtu) on 2013-02-18 15:02:14.376746. Form: Pro gradu -lomake (1 tekij\u00e4) (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake-1-tekijae). JyX data:", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2013-02-18T15:02:14Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201302181238.pdf: 1076110 bytes, checksum: 18e11ef90c50f10f1d2b32722ff5808a (MD5)\nlicense.html: 4973 bytes, checksum: c4b4493c1309a1a07d636bf4fcfa4a61 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2013-02-18T15:02:14Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201302181238.pdf: 1076110 bytes, checksum: 18e11ef90c50f10f1d2b32722ff5808a (MD5)\nlicense.html: 4973 bytes, checksum: c4b4493c1309a1a07d636bf4fcfa4a61 (MD5)\n Previous issue date: 2012", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "49 sivua", "language": null, "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": "swe", "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": "sf\u00e4risk geometri", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kartprojektioner", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stereografiska projektion", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Mercators projektion", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Albers projektion", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Sf\u00e4risk geometri och kartprojektion", "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-201302181238", "language": null, "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": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2013-02-18T15:02:15Z", "language": null, "element": "date", "qualifier": "updated", "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.accesslevel", "value": "openAccess", "language": "fi", "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": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "karttaprojektiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "projektio", "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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