{"id":3790,"date":"2024-09-22T10:44:59","date_gmt":"2024-09-22T10:44:59","guid":{"rendered":"https:\/\/flatarmal.is\/?p=3790"},"modified":"2024-09-29T18:04:36","modified_gmt":"2024-09-29T18:04:36","slug":"soguhornid-keilusnid","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=3790","title":{"rendered":"S\u00d6GUHORNI\u00d0: Keilusni\u00f0"},"content":{"rendered":"\n

Krist\u00edn Bjarnad\u00f3ttir.<\/strong><\/p>\n\n\n\n

Keila er heillandi form. Listamenn hafa gl\u00edmt vi\u00f0 a\u00f0 fanga keilulaga fj\u00f6ll \u00ed m\u00e1lverkum og lj\u00f3\u00f0um. Eldfj\u00f6ll, sem gj\u00f3sa einu sinni \u00feunnflj\u00f3tandi hrauni sem streymir jafnt ni\u00f0ur til allra hli\u00f0a, geta or\u00f0i\u00f0 keilulaga. Eldfj\u00f6ll me\u00f0 keilulagi nefnast dyngjur.<\/p>\n\n\n\n

Mynd 1 s\u00fdnir fjalli\u00f0 Skjaldbrei\u00f0i, n\u00e1nast fullkomlega keilulaga. Tali\u00f0 er a\u00f0 Skjaldbrei\u00f0ur hafi or\u00f0i\u00f0 til vi\u00f0 eldgos fyrir um 9000 \u00e1rum s\u00ed\u00f0an.<\/p>\n\n\n\n

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Mynd 1: Skjaldbrei\u00f0ur teygir sig upp \u00ed 1.066 m h\u00e6\u00f0 yfir sj\u00e1varm\u00e1l. Lj\u00f3smynd: Reg\u00edna Fann\u00fd Gu\u00f0mundsd\u00f3ttir<\/em><\/p>\n<\/div>\n\n\n\n

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Sk\u00e1ldi\u00f0 og n\u00e1tt\u00farufr\u00e6\u00f0ingurinn
J\u00f3nas Hallgr\u00edmsson orti
um Skjaldbrei\u00f0i:<\/p>\n\n\n\n

Fanna skautar faldi h\u00e1um
fjalli\u00f0, allra h\u00e6\u00f0a val,
hrauna veitir b\u00e1rum bl\u00e1um
brei\u00f0an fram um hei\u00f0ardal.<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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R\u00e9tt keila<\/strong><\/p>\n\n\n\n

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Mynd af r\u00e9ttri keilu kemur fram vi\u00f0 a\u00f0 sn\u00faa r\u00e9tthyrndum \u00fer\u00edhyrningi heilan hring um a\u00f0ra skammhli\u00f0ina (sj\u00e1 Mynd 2). Sn\u00fanings\u00e1sinn er \u00f6nnur skammhli\u00f0 \u00fer\u00edhyrningsins, og ver\u00f0ur \u00e1s keilunnar. Hornr\u00e9tt \u00e1 \u00e1sinn markast hringlaga botn.<\/p>\n<\/div>\n\n\n\n

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Mynd 2: R\u00e9tt keila<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Keilusni\u00f0: Hringur, sporbaugur, fleygbogi og brei\u00f0bogi<\/strong><\/p>\n\n\n\n

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Mynd 3: Fr\u00e1 vinstri: (1) fleygbogi, (2) hringur, (3) sporbaugur, (4) brei\u00f0bogi<\/em><\/p>\n\n\n\n

Mynd 3 s\u00fdnir hvernig fagurlega mynda\u00f0ir skur\u00f0ferlar koma fram \u00feegar keila er skorin l\u00e1r\u00e9tt, l\u00f3\u00f0r\u00e9tt e\u00f0a \u00e1 sk\u00e1. S\u00e9 skori\u00f0 l\u00e1r\u00e9tt hornr\u00e9tt \u00e1 \u00e1s keilu ver\u00f0ur skur\u00f0ferillinn hringur<\/strong> (Mynd 3 (2)).<\/p>\n\n\n\n

S\u00e9 ekki skori\u00f0 hornr\u00e9tt \u00e1 \u00e1sinn koma fram a\u00f0rir ferlar. Ef halli skur\u00f0flatarins er minni en hli\u00f0arhalli keilunnar ver\u00f0ur br\u00fan skur\u00f0flatarins eins konar teyg\u00f0ur hringur, sporbaugu<\/strong>r (Mynd 3 (3)).<\/p>\n\n\n\n

Ef halli skur\u00f0flatarins er sams\u00ed\u00f0a hli\u00f0 keilunnar, ver\u00f0ur br\u00fan skur\u00f0flatarins bogadreginn ferill sem lokast ekki, heldur v\u00edkkar eftir \u00fev\u00ed sem lengra er skori\u00f0. Ferillinn nefnist fleygbogi<\/strong> (Mynd 3 (1)).<\/p>\n\n\n\n

S\u00e9 keilan skorin \u00feannig a\u00f0 skur\u00f0fl\u00f6turinn ver\u00f0i sams\u00ed\u00f0a \u00e1s keilunnar, ver\u00f0ur skur\u00f0ferillinn enn v\u00ed\u00f0ari en \u00e1 fleygboga. Ferillinn nefnist brei\u00f0bogi<\/strong> (Mynd 3 (4)). Venjulega er fjalla\u00f0 um brei\u00f0boga sem tvo a\u00f0skilda ferla. \u00de\u00e1 er keila framlengd me\u00f0 annarri keilu sem er speglun hennar \u00ed sl\u00e9ttu \u00ed gegnum oddpunkt keilanna, hornr\u00e9ttri \u00e1 \u00e1sa \u00feeirra.<\/p>\n\n\n\n

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Heiti keilusni\u00f0anna<\/strong><\/p>\n\n\n\n

Ferlarnir sem h\u00e9r hafa veri\u00f0 nefndir, hringur<\/strong>, sporbaugur<\/strong>, fleygbogi<\/strong> og brei\u00f0bogi<\/strong>, hafa veri\u00f0 vi\u00f0f\u00f6ng st\u00e6r\u00f0fr\u00e6\u00f0inga fr\u00e1 forn\u00f6ld, allt fr\u00e1 t\u00edmum Forn-Grikkja \u00e1 fj\u00f3r\u00f0u \u00f6ld fyrir Krist. St\u00e6r\u00f0fr\u00e6\u00f0ingnum Menaechmus er eigna\u00f0ur hei\u00f0urinn af \u00fev\u00ed a\u00f0 hafa uppg\u00f6tva\u00f0 keilusni\u00f0 um 360-350 f.Kr. Heiti keilusni\u00f0a \u00e1 \u00f6\u00f0rum evr\u00f3pskum tungum\u00e1lum eru rakin til forngr\u00edsku:<\/p>\n\n\n\n

\u00cdslenska<\/th>Danska<\/th>Enska<\/th>\u00de\u00fdska<\/th>Franska<\/th><\/tr><\/thead>
hringur<\/td>cirkel<\/td>circle<\/td>Ring<\/td>cercle<\/td><\/tr>
sporbaugur<\/td>ellipse<\/td>ellipse<\/td>Ellipse<\/td>ellipse<\/td><\/tr>
fleygbogi<\/td>parabel<\/td>parabola<\/td>Parabel<\/td>parabole<\/td><\/tr>
brei\u00f0bogi<\/td>hyperbel <\/td>hyperbola<\/td>Hyperbel<\/td>hyperbole<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Heiti keilusni\u00f0a eru mj\u00f6g s\u00e9rt\u00e6k or\u00f0 og koma sjaldan fyrir utan st\u00e6r\u00f0fr\u00e6\u00f0i, e\u00f0lisfr\u00e6\u00f0i og stj\u00f6rnufr\u00e6\u00f0i. Hva\u00f0 var\u00f0 til \u00feess a\u00f0 ferlarnir hlutu n\u00fd \u00f3skyld heiti \u00e1 \u00edslensku?<\/p>\n\n\n\n

\u00c1ri\u00f0 1842 kom \u00fat \u00edslensk \u00fe\u00fd\u00f0ing \u00e1 b\u00f3kinni Astronomi<\/em> eftir danska pr\u00f3fessorinn Georg Frederik Ursin (1830), undir heitinu Stj\u00f6rnufr\u00e6\u00f0i, ljett og handa al\u00fe\u00ed\u00f0u<\/em>. \u00de\u00fd\u00f0andinn var J\u00f3nas Hallgr\u00edmsson, sk\u00e1ld og n\u00e1tt\u00farufr\u00e6\u00f0ingur. \u00de\u00fd\u00f0ingar J\u00f3nasar \u00e1 heitum keilusni\u00f0anna eru hlj\u00f3mf\u00f6gur or\u00f0. Ein \u00e1st\u00e6\u00f0a \u00feess a\u00f0 hann valdi a\u00f0 \u00fe\u00fd\u00f0a or\u00f0in g\u00e6ti veri\u00f0 s\u00fa a\u00f0 \u00e1hersla er sjaldnast \u00e1 fyrsta atkv\u00e6\u00f0i \u00ed erlendu or\u00f0unum. Or\u00f0in ellipsa<\/strong>, parab\u00f3la<\/strong> og hyperb\u00f3la<\/strong> hafa \u00fe\u00f3 oft veri\u00f0 notu\u00f0 sem t\u00f6kuor\u00f0 \u00ed \u00edslensku enda voru kennslub\u00e6kur \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i \u00ed framhaldssk\u00f3lum gjarnan \u00e1 nor\u00f0urlandam\u00e1lum langt fram \u00e1 tuttugustu \u00f6ld. \u00deegar fari\u00f0 var a\u00f0 \u00fe\u00fd\u00f0a og semja kennslub\u00e6kur \u00e1 \u00edslensku um \u00feessi efni, voru heiti J\u00f3nasar tekin upp \u00ed kennslub\u00f3kunum.<\/p>\n\n\n\n

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Keilusni\u00f0 og himinhnettirnir<\/strong><\/p>\n\n\n\n

Hva\u00f0a erindi eiga \u00feessir ferlar \u00ed b\u00f3k um stj\u00f6rnufr\u00e6\u00f0i? Allt fr\u00e1 forn\u00f6ld gl\u00edmdu menn vi\u00f0 a\u00f0 \u00e1tta sig \u00e1 gangi himintunglanna. Lengi var tali\u00f0 a\u00f0 allir himinhnettir, s.s. s\u00f3l, tungl, reikistj\u00f6rnurnar Venus, Mars og J\u00fap\u00edter \u00e1samt fastastj\u00f6rnunum, snerust um j\u00f6r\u00f0ina. \u00dea\u00f0 nefndist jar\u00f0mi\u00f0jukenning<\/strong>. Samkv\u00e6mt henni virtust brautir hnattanna afar fl\u00f3knar, en \u00feekking \u00e1 \u00feeim n\u00fdttist \u00fe\u00f3 til a\u00f0 segja fyrir um gang hnattanna og hvar og hven\u00e6r \u00feeirra v\u00e6ri von. \u00c1 sext\u00e1ndu \u00f6ld hlaut s\u00f3lmi\u00f0jukenningin<\/strong> um a\u00f0 s\u00f3lin v\u00e6ri mi\u00f0ja s\u00f3lkerfisins vaxandi fylgi fyrir verk \u00feeirra K\u00f3pern\u00edkusar (1473\u20131543), Keplers (1571\u20131630) og Galileos Galilei (1564\u20131642), og brautir hnattanna einf\u00f6ldu\u00f0ust til muna.<\/p>\n\n\n\n

Fyrst var tali\u00f0 a\u00f0 j\u00f6r\u00f0in gengi \u00ed hring \u00ed kringum um s\u00f3lu, en s\u00ed\u00f0ar var s\u00fdnt fram \u00e1 a\u00f0 braut jar\u00f0arinnar v\u00e6ri sporbaugur og s\u00f3lin v\u00e6ri \u00ed \u00f6\u00f0rum brennipunkti hans. Sama m\u00e1li gegnir um a\u00f0rar reikistj\u00f6rnur. Brautir reikistjarnanna eins og jar\u00f0arinnar ver\u00f0a til fyrir a\u00f0dr\u00e1ttarafli\u00f0. Milli jar\u00f0ar og s\u00f3lar er gagnkv\u00e6mt a\u00f0dr\u00e1ttarafl sem heldur litlum massa jar\u00f0arinnar og hinna reikistjarnanna \u00e1 fer\u00f0 eftir sporbaugum umhverfis st\u00f3ran massa s\u00f3larinnar.<\/p>\n\n\n\n

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Hnitakerfi\u00f0<\/strong><\/p>\n\n\n\n

Kartes\u00edskt hnitakerfi hefur reynst heppilegur rammi til a\u00f0 setja keilusni\u00f0sferlana skilmerkilega fram. Or\u00f0i\u00f0 \u201ekartes\u00edskt\u201c v\u00edsar til heimspekingsins og st\u00e6r\u00f0fr\u00e6\u00f0ingsins Ren\u00e9 Descartes (1596\u20131650) sem talinn er upphafsma\u00f0ur tv\u00edv\u00ed\u00f0s r\u00e9tthyrnds hnitakerfis. S\u00e9rhver punktur \u00ed sl\u00e9ttunni er settur fram me\u00f0 tveimur t\u00f6lum, hnitum. T\u00f6lurnar t\u00e1kna annars vegar fjarl\u00e6g\u00f0 fr\u00e1 l\u00f3\u00f0r\u00e9ttri talnal\u00ednu, en hins vegar fr\u00e1 l\u00e1r\u00e9ttri talnal\u00ednu. Me\u00f0 \u00fev\u00ed a\u00f0 f\u00e6ra r\u00famfr\u00e6\u00f0i inn \u00ed hnitakerfi var\u00f0 unnt a\u00f0 tengja saman algebru og r\u00famfr\u00e6\u00f0i og setja r\u00famfr\u00e6\u00f0ileg fyrirbrig\u00f0i fram me\u00f0 algebru.<\/p>\n\n\n\n

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Hringur<\/strong><\/p>\n\n\n\n

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Jafna hrings me\u00f0 mi\u00f0ju \u00ed upphafspunkti hnitakerfisins, (0, 0), er x2<\/sup> + y2<\/sup> = r2<\/sup>, \u00fear sem r er rad\u00edi<\/strong> e\u00f0a geisli<\/strong> hringsins, sj\u00e1 Mynd 4. J\u00f6fnu hrings m\u00e1 l\u00edka rita: <\/p>\n\n\n

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Mynd 4: Hringur<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Sporbaugur<\/strong><\/p>\n\n\n\n

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Sporbaugur<\/strong> er eins konar teyg\u00f0ur hringur og jafna sporbaugs er n\u00e1skyld j\u00f6fnu hrings. \u00cd sporbaug eru tveir h\u00e1lf\u00e1sar, a og b, \u00ed sta\u00f0 geislans r. Lang\u00e1s<\/strong> hefur lengdina 2a, en skamm\u00e1s<\/strong> lengdina 2b, sj\u00e1 Mynd 5. Jafna sporbaugs me\u00f0 mi\u00f0ju \u00ed (0, 0) er: <\/p>\n\n\n

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Mynd 5: Sporbaugur<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Brei\u00f0bogi<\/strong><\/p>\n\n\n\n

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\u00de\u00f3tt ferlarnir sporbaugur og brei\u00f0bogi s\u00e9u \u00f3l\u00edkir a\u00f0 sj\u00e1 eru j\u00f6fnur \u00feeirra svipa\u00f0ar. J\u00f6fnur brei\u00f0boga<\/strong> me\u00f0 mi\u00f0ju \u00ed (0, 0) eru:<\/p>\n\n\n

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\u00fear sem a er l\u00e1r\u00e9tt fjarl\u00e6g\u00f0 oddpunkta brei\u00f0bogans A fr\u00e1 upphafspunkti og b er l\u00f3\u00f0r\u00e9tt fjarl\u00e6g\u00f0 oddpunkta systurferilsins B fr\u00e1 upphafspunkti, sj\u00e1 Mynd 6.  <\/p>\n<\/div>\n\n\n\n

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Mynd 6: Brei\u00f0bogi<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

Eiginleiki brei\u00f0boga er a\u00f0 n\u00e1lgast beinar l\u00ednur, sem nefnast a\u00f0fellur <\/strong>(e. asymtotes), \u00e6 meir eftir \u00fev\u00ed sem fj\u00e6r dregur oddpunktum brei\u00f0bogans. J\u00f6fnur a\u00f0fellanna eru <\/p>\n\n\n

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Fleygbogi<\/strong><\/p>\n\n\n\n

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Jafna fleygboga<\/strong> er y = ax2<\/sup>, ef oddpunktur hans er \u00ed upphafspunktinum (0, 0). Sj\u00e1 Mynd 7.<\/p>\n<\/div>\n\n\n\n

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Mynd 7: Fleygbogi<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Algebra keilusni\u00f0a<\/strong><\/p>\n\n\n\n

\u00deegar grannt er sko\u00f0a\u00f0 eru l\u00edkindi me\u00f0 j\u00f6fnum allra keilusni\u00f0anna. \u00d6nnur hvor breytan x e\u00f0a y e\u00f0a b\u00e1\u00f0ar koma alltaf fyrir \u00ed \u00f6\u00f0ru veldi. Almennt gildir jafnan:<\/p>\n\n\n\n

ax2<\/sup> + bxy + cy2<\/sup> + dx + ey + f = 0<\/p>\n\n\n\n

fyrir \u00f6ll keilusni\u00f0in ef stu\u00f0larnir a, b, og c eru ekki allir n\u00fall samt\u00edmis.<\/p>\n\n\n\n

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Brennipunktur, st\u00fdril\u00edna og hringvik<\/strong><\/p>\n\n\n\n

Keilusni\u00f0in sporbaugur, fleygbogi og brei\u00f0bogi hafa brennipunkt<\/strong> (e. focus), einn e\u00f0a tvo, og st\u00fdril\u00ednu<\/strong> (e. directrix). Hringvik<\/strong>, (e. excentricity) er hlutfall fjarl\u00e6g\u00f0ar punkts \u00e1 keilusni\u00f0i fr\u00e1 brennipunkti keilusni\u00f0sins og fjarl\u00e6g\u00f0ar hans fr\u00e1 st\u00fdril\u00ednunni.<\/p>\n\n\n\n

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Fleygbogi<\/strong><\/p>\n\n\n\n

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Ef oddpunktur fleygboga er \u00ed (0,0) og brennipunkturinn<\/strong> hefur hnitin (0, p) er jafna st\u00fdril\u00ednu<\/strong> y = \u2013 p, og jafna fleygbogans<\/strong>:<\/p>\n\n\n

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Um s\u00e9rhvern punkt M(x, y) \u00e1 fleygboganum gildir a\u00f0 fjarl\u00e6g\u00f0in MB stendur \u00ed hlutfalli vi\u00f0 fjarl\u00e6g\u00f0ina MD \u00fear sem D(x, \u2013 p) er punktur \u00e1 st\u00fdril\u00ednunni. \u00c1 mynd 8 er MB = MD og hringviki\u00f0<\/strong> MB\/MD=1<\/p>\n\n\n\n

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Mynd 8: Fleygbogi, brennipunktur og st\u00fdril\u00edna<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Sporbaugur<\/strong><\/p>\n\n\n\n

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Sporbaugur hefur tvo brennipunkta<\/strong>. Ef mi\u00f0ja sporbaugsins er \u00ed (0, 0) eru brennipunktarnir \u00ed (c, 0) og (-c, 0).  <\/p>\n\n\n\n

Ef a > b gildir um c jafnan: c2<\/sup> = a2<\/sup> \u2013 b2<\/sup>.  <\/p>\n\n\n\n

J\u00f6fnur st\u00fdril\u00edna<\/strong> l\u00e1r\u00e9tts sporbaugs eru:<\/p>\n\n\n

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Mynd 9: Sporbaugur, brennipunktar og st\u00fdril\u00ednur<\/em><\/p>\n\n\n\n

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Summa fjarl\u00e6g\u00f0a punkts \u00e1 sporbaug fr\u00e1 brennipunktunum tveimur er fasti:<\/p>\n\n\n\n

PF1<\/sub> + PF2<\/sub> = 2a<\/p>\n\n\n\n

\u00fear sem a er h\u00e1lf lengd lang\u00e1ss sporbaugsins.<\/p>\n\n\n\n

L\u00edta m\u00e1 \u00e1 hring<\/strong> sem s\u00e9rtilvik af sporbaug. Brennipunktur hrings fellur saman vi\u00f0 mi\u00f0ju hans.<\/p>\n<\/div>\n\n\n\n

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Mynd 10: Sporbaugur, fjarl\u00e6g\u00f0 punkts fr\u00e1 brennipunktum<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n
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Brei\u00f0bogi<\/strong><\/p>\n\n\n\n

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Topppunktar brei\u00f0boga me\u00f0 mi\u00f0ju \u00ed (0, 0) er \u00ed (a, 0) og (-a,0) \u00fear sem a er h\u00e1lf lengd lang\u00e1ss brei\u00f0bogans, sj\u00e1 Mynd 11.<\/p>\n\n\n\n

Brei\u00f0bogi hefur l\u00edka l\u00f3\u00f0r\u00e9ttan \u00e1s, b, sj\u00e1 Mynd 6.  <\/p>\n\n\n\n

Brennipunktar brei\u00f0boga me\u00f0 l\u00e1r\u00e9ttan \u00e1s eru F1 <\/sub>= (c, 0) og F2<\/sub> = (-c, 0), \u00fear sem a2<\/sup> + b2<\/sup> = c2<\/sup>.  <\/p>\n\n\n\n

J\u00f6fnur st\u00fdril\u00edna l\u00e1r\u00e9tts brei\u00f0boga eru \u00fe\u00e1:<\/p>\n\n\n

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\"\"<\/figure>\n<\/div><\/div>\n\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

Mynd 11: Brei\u00f0bogi, brennipunktar og st\u00fdril\u00ednur<\/em><\/p>\n\n\n\n

<\/div>\n<\/div>\n<\/div>\n\n\n\n
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Mismunur fjarl\u00e6g\u00f0a punkts \u00e1 brei\u00f0boga fr\u00e1 brennipunktunum tveimur er fasti:<\/p>\n\n\n\n

\u2502\u2502PF2<\/sub> \u2502\u2013 \u2502PF1 <\/sub>\u2502\u2502= 2a<\/p>\n<\/div>\n\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

Mynd 12: Brei\u00f0bogi, mismunur fjarl\u00e6g\u00f0a punkts fr\u00e1 brennipunktunum<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n
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Hringvik<\/strong><\/p>\n\n\n\n

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\n

Hringvik<\/strong>, e, er eins og \u00e1\u00f0ur sag\u00f0i hlutfalli\u00f0 milli fjarl\u00e6g\u00f0ar punkts fr\u00e1 brennipunkti ferils og st\u00fdril\u00ednu ferilsins, sj\u00e1 Mynd 13.<\/p>\n\n\n\n

Hringvik sporbaugs er minna en 1. Hringvik fleygboga er 1 e\u00f0a st\u00e6rra.<\/p>\n\n\n\n

\u00deannig m\u00e1 l\u00edta \u00e1 a\u00f0 hringvik beinnar l\u00ednu s\u00e9 \u00f3endanlegt.<\/p>\n<\/div>\n\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

Mynd 13: Hringvik<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n
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\u00ddmsir eiginleikar keilusni\u00f0a<\/strong><\/p>\n\n\n\n

Brautir margra himinhnatta eru sporbaugar<\/strong> utan um a\u00f0ra \u00feyngri hnetti sem eru \u00ed \u00f6\u00f0rum brennipunkti sporbaugsins.<\/p>\n\n\n\n

Ferlar sumra halastjarna eru brei\u00f0bogar<\/strong>.<\/p>\n\n\n\n

Kastferlar hluta \u00ed \u00feyngdarsvi\u00f0i eru fleygbogar<\/strong> ef horft er framhj\u00e1 loftm\u00f3tst\u00f6\u00f0u.<\/p>\n\n\n\n

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\n

Fleygbogi<\/strong> hefur \u00feann eiginleika a\u00f0 s\u00e9 geislum varpa\u00f0 a\u00f0 honum sams\u00ed\u00f0a \u00e1s hans endurvarpast \u00feeir \u00ed brennipunktinn.<\/p>\n\n\n\n

S\u00e9 lj\u00f3s kveikt \u00ed brennipunkti spegils me\u00f0 fleygbogalaga\u00f0 \u00feversni\u00f0, sj\u00e1 Mynd 14, varpast lj\u00f3sgeislarnir \u00fat sams\u00ed\u00f0a \u00e1s fleygbogans. \u00deessi eiginleiki er n\u00fdttur vi\u00f0 ger\u00f0 lj\u00f3skastara.<\/p>\n\n\n\n

Eiginleikinn er l\u00edka nota\u00f0ur \u00ed st\u00e6kkunarglerjum (safnglerjum). \u00de\u00e1 er sams\u00ed\u00f0a geislum lj\u00f3sgjafa, til d\u00e6mis s\u00f3lar, safna\u00f0 \u00ed brennipunkt. S\u00e9 lj\u00f3sgjafinn s\u00f3lin m\u00e1 h\u00e6glega nota safngler til \u00edkveikju.<\/p>\n<\/div>\n\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

Mynd 14: Sams\u00ed\u00f0a geislarnir Q1<\/sub>P1<\/sub>, Q2<\/sub>P2<\/sub>, Q3<\/sub>P3<\/sub>\u2026 endurvarpast \u00ed fleygbogal\u00f6gu\u00f0um ferli og safnast saman \u00ed brennipunktinum F. Einnig g\u00e6tu geislar fr\u00e1 brennipunktinum F endurvarpast sem sams\u00ed\u00f0a geislar. <\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n
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Flatarm\u00e1l flata innan keilusni\u00f0a<\/strong><\/p>\n\n\n\n

Hringur<\/strong><\/p>\n\n\n\n

Flatarm\u00e1l hrings er F = \u03c0r2<\/sup><\/p>\n\n\n\n

Sporaskja<\/strong><\/p>\n\n\n\n

Fl\u00f6turinn innan sporbaugs nefnist sporaskja. Flatarm\u00e1l spor\u00f6skju er F = \u03c0ab.<\/p>\n\n\n\n

A\u00f0rir ferlar eru opnir. H\u00e6gt er a\u00f0 reikna flatarm\u00e1l sv\u00e6\u00f0a sem afmarkast af fleygboga og beinni l\u00ednu e\u00f0a brei\u00f0boga og annars ferils.<\/p>\n\n\n\n

\n\n\n\n

\u00c6fing<\/strong><\/p>\n\n\n\n

Festi\u00f0 spotta me\u00f0 teiknib\u00f3lum \u00ed b\u00e1\u00f0a enda \u00e1 spjald \u00feannig a\u00f0 slaki s\u00e9 \u00e1 spottanum. Taki\u00f0 bl\u00fdant og spenni\u00f0 spottann og renni\u00f0 s\u00ed\u00f0an bl\u00fdantinum eins langt og komist ver\u00f0ur b\u00e1\u00f0um megin vi\u00f0 tengil\u00ednu teiknib\u00f3lanna. Samkv\u00e6mt Mynd 10 \u00e6tti a\u00f0 koma fram sporbaugur.  <\/p>\n\n\n\n

\n\n\n\n
\"\"<\/figure>
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Krist\u00edn Bjarnad\u00f3ttir, pr\u00f3fessor emerita vi\u00f0 H\u00e1sk\u00f3la \u00cdslands<\/em><\/p>\n\n\n\n

Bjarnhei\u00f0ur Kristinsd\u00f3ttir, lektor vi\u00f0 H\u00e1sk\u00f3la \u00cdslands vann myndir \u00ed GeoGebra<\/em><\/p>\n<\/div><\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Krist\u00edn Bjarnad\u00f3ttir. Keila er heillandi form. Listamenn hafa gl\u00edmt vi\u00f0 a\u00f0 fanga keilulaga fj\u00f6ll \u00ed m\u00e1lverkum og lj\u00f3\u00f0um. Eldfj\u00f6ll, sem gj\u00f3sa einu sinni \u00feunnflj\u00f3tandi hrauni sem streymir jafnt ni\u00f0ur til allra hli\u00f0a, geta or\u00f0i\u00f0 keilulaga. Eldfj\u00f6ll me\u00f0 keilulagi nefnast dyngjur. Mynd 1 s\u00fdnir fjalli\u00f0 Skjaldbrei\u00f0i, n\u00e1nast fullkomlega keilulaga. Tali\u00f0 er a\u00f0 Skjaldbrei\u00f0ur hafi or\u00f0i\u00f0 til…<\/p>\n","protected":false},"author":1,"featured_media":3812,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[35,39,36,42,34,30,41,38,29,32,37,33,31,40],"class_list":["post-3790","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-breidbogi","tag-brennipunktur","tag-ellipsa","tag-flatarmal-keilusnida","tag-fleygbogi","tag-hringur","tag-hringvik","tag-hyperbola","tag-keila","tag-keilusnid","tag-parabola","tag-rett-keila","tag-sporbaugur","tag-styrilina"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/3790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3790"}],"version-history":[{"count":119,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/3790\/revisions"}],"predecessor-version":[{"id":4004,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/3790\/revisions\/4004"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/3812"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}