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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Eskola, Edvin", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-06-04T07:09:23Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-06-04T07:09:23Z", "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/123456789/103052", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 pro gradu -tutkielmassa tutustutaan yleisell\u00e4 tasolla Apollonioksen ympyr\u00e4pakkauksiin, niiden olemassaoloon ja konstruointiin. Apollonioksen lauseen nojalla kolmen pareittain toisiaan sivuavan ympyr\u00e4n kolmikoille on\nolemassa kaksi sellaista ympyr\u00e4\u00e4, jotka sivuavat pareittain kaikkia kolmikon ympyr\u00f6it\u00e4. Saaduista viidest\u00e4 ympyr\u00e4st\u00e4 voidaan valita uusi pareittain toisiaan sivuavien ympyr\u00f6iden kolmikko. Soveltamalla Apollonioksen lausetta\nuusiin kolmikoihin ja jatkamalla t\u00e4llaista konstruktiota, saadaan Apollonioksen ympyr\u00e4pakkaus, jossa on \u00e4\u00e4rett\u00f6m\u00e4n monta ympyr\u00e4\u00e4.\nTutkielman aluksi todistetaan Apollonioksen lause hy\u00f6dynt\u00e4m\u00e4ll\u00e4 ympyr\u00f6iden kuvautumista inversiossa. Ennen lauseen todistamista perehdyt\u00e4\u00e4n\ninversion geometrisiin ominaisuuksiin ja aiheeseen liittyviin tuloksiin.\nT\u00e4m\u00e4n j\u00e4lkeen jatketaan pareittain toisiaan sivuavien ympyr\u00f6iden tutkimista m\u00e4\u00e4rittelem\u00e4ll\u00e4 Descartesin lause sek\u00e4 lauseen laajennettu versio,\nkompleksinen Descartesin lause. Descartesin lauseen mukaan nelj\u00e4n pareittain toisiaan sivuavan ympyr\u00e4n kaarevuudet k toteuttavat toisen asteen yht\u00e4l\u00f6n 2(k_1^2+k_2^2+k_3^2+k_4^2)=(k_1+k_2+k_3+k_4)^2. Siten jos kolmen pareittain toisiaan sivuavan ympyr\u00e4n s\u00e4teet tunnetaan, voidaan Apollonioksen lauseesta\nsaatavien kahden uuden ympyr\u00e4n s\u00e4teet selvitt\u00e4\u00e4 Descartesin lauseen avulla.\nKompleksinen Descartesin lause on Descartesin lausetta laajempi tulos, jossa toisen asteen yht\u00e4l\u00f6\u00f6n liitet\u00e4\u00e4n my\u00f6s ympyr\u00f6iden keskipisteet merkitsem\u00e4ll\u00e4 keskipisteit\u00e4 kompleksilukujen avulla. Kompleksisen Descartesin\nlauseen avulla voidaan selvitt\u00e4\u00e4 my\u00f6s Apollonioksen lauseesta saatavien ympyr\u00f6iden keskipisteet.\nTutkielman viimeisess\u00e4 osassa tutkitaan ympyr\u00e4pakkausten konstruointia Geogebra-ohjelmalla. Ympyr\u00e4pakkaus konstruoidaan kolmen annetun pisteen suhteen hy\u00f6dynt\u00e4m\u00e4ll\u00e4 tutkielmassa esitettyj\u00e4 tuloksia.", "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 2025-06-04T07:09:23Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-06-04T07:09:23Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "52", "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": "Descartesin lause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kompleksinen Descartesin lause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Apollonioksen ympyr\u00e4pakkaukset", "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-202506044857", "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"}]
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