Ellipsi

Tämä artikkeli kertoo kartioleikkauksesta, muut merkitykset sivulla Ellipsi (täsmennyssivu).

Ellipsi (suomalaisittain yleensä soikio^[1] tai joskus myös ovaali) on suljettu toisen asteen käyrä.^[1] Ellipsi on myös yksi kartioleikkauksista, niiden tason pisteiden joukko, joiden etäisyyksien summa kahdesta annetusta pisteestä on vakio.

Matemaattinen määritelmä tehdään seuraavasti. Olkoot F1 ja F2 kaksi tason kiinteätä pistettä. Ellipsi on käyrä, jolle kuuluu jokainen tason piste X, jonka F1:stä ja F2:sta mitattujen etäisyyksien summalla XF1 + XF2 on vakioarvo. Ellipsin soikeus määräytyy siitä, kuinka paljon on XF1 + XF2 suurempi kuin pisteiden F1 ja F2 välinen etäisyys.

Pisteitä F1 ja F2 sanotaan ellipsin polttopisteiksi. Suoria, joiden suhteen ellipsi on symmetrinen, sanotaan ellipsin akseleiksi. Suoraa AB kutsutaan ellipsin isoakseliksi. Jana a on isoakselin puolikas. Suoraa CD kutsutaan ellipsin pikkuakseliksi. Jana b on pikkuakselin puolikas.

Pinta-ala

Ellipsin pinta-ala $A$ saadaan kaavasta

A=\pi \cdot ab,

missä a ja b ovat ellipsin puoliakseleita.

Kaavasta voidaan huomata, että erityistapauksessa, jossa puoliakselit ovat yhtä pitkiä, kuvio on ympyrä ja pinta-alan lausekkeeksi tulee π·r².

Ellipsin kehän pituutta $p$ ei voi alkeisfunktioiden avulla lausua suljetussa muodossa. Tarkka kaava on

$p=4\ a\int _{0}^{\pi /2}{\sqrt {1-\epsilon ^{2}\sin ^{2}{t}}}\;dt,$

jossa $\epsilon$ on ellipsin eksentrisyys. Se sisältää toisen lajin elliptisen integraalin.

Ellipsin yhtälö

Kun ellipsin keskipiste on pisteessä (x₀,y₀), on sen yhtälö muotoa

{\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}=1\!

, jossa

a,b\in \mathbb {R}

Ellipsin yhtälö parametrimuodossa:

\left\{{\begin{matrix}x=x_{0}+a\cos {t}\\y=y_{0}+b\sin {t}\end{matrix}}\right.

, jossa

a,b,t,x_{0},y_{0}\in \mathbb {R}

Ellipsin yhtälö voidaan myös esittää muodossa

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\!

, jossa

A,B,C,D,E,F\in \mathbb {R}

Kaavoissa a on x-akselin suuntaisen puoliakselin pituus ja b y-akselin suuntaisen puoliakselin pituus.

Jos a = b = r, kyseessä on ympyrä, jonka säde on r.

Katso myös

Ellipsoidi

Lähteet

↑ ^a ^b ellipsi. Kielitoimiston sanakirja. Helsinki: Kotimaisten kielten keskus, 2024.

Aiheesta muualla

Kuvia tai muita tiedostoja aiheesta Ellipsi Wikimedia Commonsissa

Kirjallisuutta

Kivelä, Simo K.: Algebra ja geometria. Espoo: Otatieto, 1989. ISBN 951-672-103-6
Rikkonen, Harri: Matematiikan pitkä peruskurssi I – Vektorialgebra ja analyyttinen geometria. Helsinki: Otakustantamo, 1969. ISBN 951-671-067-0
Pitkäranta, Juhani: Calculus Fennicus – TKK:n 1. lukuvuoden laaja matematiikka (2000–2013) (pdf) Helsinki: Avoimet oppimateriaalit ry. ISBN 978-952-7010-12-9 ISBN 978-952-7010-6 (pdf). Viitattu 8.7.2019.

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