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[{"key": "dc.contributor.advisor", "value": "\u00c4kkinen, Tuomo", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hovinm\u00e4ki, Noora", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-04-07T07:56:21Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-04-07T07:56:21Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/68478", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on avata matemaattisen tasobiljardin keskeisi\u00e4 tuloksia geometrian keinoin. Lis\u00e4ksi tavoitteena on muodostaa lukioik\u00e4isille suunnattu biljardigeometriaan liittyv\u00e4 teht\u00e4v\u00e4kokonaisuus, jonka avulla opiskelijat p\u00e4\u00e4sisiv\u00e4t tutustumaan matemaattiseen biljardiin. Tutkielmassa keskityt\u00e4\u00e4n p\u00e4\u00e4sasiassa ympyr\u00e4- ja ellipsibiljardiin ja tutustutaan hieman my\u00f6s monikulmiobiljardiin neli\u00f6- ja kolmiobiljardin muodossa.\nTutkielmassa k\u00e4yd\u00e4\u00e4n ensin l\u00e4pi konveksin, suljetun ja rajoitetun tasoalueen yleinen tasobiljardikuvaus, josta johdetaan my\u00f6hemmin ympyr\u00e4biljardikuvaus. Ympyr\u00e4biljardissa huomataan, ett\u00e4 kiertokulmaa vastaavan kaaren pituus m\u00e4\u00e4r\u00e4\u00e4 suoraan sen, onko ratak\u00e4yr\u00e4 jaksollinen vai ei. Lis\u00e4ksi havaitaan, ett\u00e4 ympyr\u00e4biljardissa rata kulkee aina joko edestakaisin ympyr\u00e4n keskipisteen kautta tai kiert\u00e4en keskipistett\u00e4 sellaisia ratasegmenttej\u00e4 pitkin, jotka sivuavat sisemp\u00e4\u00e4 samakeskist\u00e4 ympyr\u00e4\u00e4. T\u00e4t\u00e4 ratak\u00e4yrien sivuamaa ympyr\u00e4\u00e4 kutsutaan ympyr\u00e4biljardin kaustiseksi k\u00e4yr\u00e4ksi.\nYmpyr\u00e4biljardin j\u00e4lkeen siirryt\u00e4\u00e4n neli\u00f6biljardiin, miss\u00e4 huomataan olevan varsin paljon samankaltaisuutta ympyr\u00e4biljardin kanssa. Neli\u00f6biljardissa biljardipallon l\u00e4ht\u00f6kulman avulla pystyst\u00e4\u00e4n m\u00e4\u00e4ritt\u00e4m\u00e4\u00e4n t\u00e4ysin se, onko biljardirata neli\u00f6ll\u00e4 jaksollinen vai ei. Neli\u00f6n jaksottomien ratak\u00e4yrien yhteydess\u00e4 tutustutaan my\u00f6s kvasijaksollisuuden k\u00e4sitteeseen.\nKolmiobiljardia tutkittaessa huomataan, ett\u00e4 ratak\u00e4yr\u00e4n k\u00e4ytt\u00e4ytyminen vaihtelee melko suuresti sen mukaan, onko biljardip\u00f6yt\u00e4 ter\u00e4v\u00e4kulmaisen, suorakulmaisen vai tylpp\u00e4kulmaisen kolmion muotoinen. Ter\u00e4v\u00e4kulmaisten kolmioiden tapauksessa l\u00f6ydet\u00e4\u00e4n aina 3-jaksollinen Fagnanon ratak\u00e4yr\u00e4. Suorakulmaisilla kolmioilla taas voidaan muodostaa mink\u00e4 tahansa kolmion sis\u00e4pisteen kautta jaksollinen ratak\u00e4yr\u00e4. Sen sijaan tylpp\u00e4kulmaisten kolmioiden tapauksissa ei edes tiedet\u00e4, onko jokaisella kolmiolla jaksollista biljardirataa. Kolmion jaksollisten ratak\u00e4yrien yhteydess\u00e4 perehdyt\u00e4\u00e4n my\u00f6s muun muassa kohtisuoriin ratak\u00e4yriin, S-ratak\u00e4yriin ja peiliratak\u00e4yriin.\nViimeisen\u00e4 biljardia tarkastellaan elliptisell\u00e4 biljardip\u00f6yd\u00e4ll\u00e4, miss\u00e4 ratak\u00e4yr\u00e4n muotoon n\u00e4ytt\u00e4\u00e4 vaikuttavan suuresti se, kulkeeko biljardirata ellipsin polttopisteiden kautta, niiden v\u00e4list\u00e4 vai niiden ulkopuolelta. Polttopisteiden kautta kulkeva ratak\u00e4yr\u00e4 suppenee kohti isoakselia, kun taas polttopisteiden v\u00e4list\u00e4 tai niiden ulkopuolelta kulkeville radoille muodostuu hyperbelin tai ellipsin muotoinen kaustinen k\u00e4yr\u00e4. Lis\u00e4ksi Ponceletin lauseen sis\u00e4lt\u00f6 osoittautuu varsin hy\u00f6dylliseksi ellipsin jaksollisia ratak\u00e4yri\u00e4 etsitt\u00e4ess\u00e4.\nTutkielman loppuun on koottu lukioik\u00e4isille suunnattu teht\u00e4v\u00e4kokonaisuus biljardigeometriasta malliratkaisuineen ja se sis\u00e4lt\u00e4\u00e4 vaihtelevasti laskemista, v\u00e4itteiden todeksi osoittamista ja asioiden tutkimista GeoGebran avulla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2020-04-07T07:56:21Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-04-07T07:56:21Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "61", "language": "", "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": "biljardiradat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ympyr\u00e4biljardi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ellipsibiljardi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "monikulmiobiljardi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "neli\u00f6biljardi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kolmiobiljardi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matemaattista biljardia ja geometriaa", "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-202004072693", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "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": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "biljardi", "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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