Ka Hoʻohālikelike Ellipse ma ke Geometry
He piʻo koʻikoʻi ka ellipse i ke geometry, e ʻike ʻia ana ma nā ʻano like ʻole, mai ka makemakika maʻemaʻe a hiki i nā noi ma ka physics, ka ʻenekinia, a me ka astronomy. I ka ʻōlelo maʻalahi, hiki ke hoʻomaopopo ʻia kahi ellipse ma ke ʻano he "pōʻai i hohola ʻia" i lilo ia i mea lōʻihi i hoʻokahi ʻaoʻao. Eia nō naʻe, ʻoi aku ka hoihoi o ka wehewehe kūhelu o kahi ellipse: ʻo ka ellipse ka hoʻonohonoho o nā kiko āpau i loko o kahi mokulele nona ka huina o ko lākou mamao mai ʻelua mau kiko paʻa (i kapa ʻia he foci) e mau mau ana. Mai kēia wehewehe ʻana, hiki ke loaʻa a aʻo ʻia ke kaulike o kahi ellipse, ma nā ʻano maʻamau a me nā ʻano laulā.
1. Ke hoʻomaopopo nei i nā Ellipses a me kā lākou mau Elements
I mea e hoʻomaopopo ai i ke ʻano o ka ellipse, pono mākou e ʻike i nā kumu nui o kahi ellipse:
1. Kikowaena o ka ellipse (kikowaena): ke kiko waena o ka ellipse, i hōʻailona pinepine ʻia \((h, k)\).
2. ʻO ke axis nui: ke anawaena lōʻihi loa o ka ellipse.
3. ʻO ke koʻi liʻiliʻi: ke anawaena pōkole loa o ka ellipse e kū pololei ana i ke koʻi nui.
4. Kūleʻa (foci): ʻelua mau kiko paʻa e lawelawe ana ma ke ʻano he kuhikuhi no ka wehewehe ʻana o kahi ellipse, i hōʻike pinepine ʻia ʻo \(F_1\) a me \(F_2\).
5. Radius semimajor: hapalua o ka lōʻihi o ke axis nui, i hōʻailona ʻia ʻo \(a\).
6. Ka radius semiminor: ka hapalua o ka lōʻihi o ke axis liʻiliʻi, i hōʻike ʻia ʻo \(b\).
7. Ka mamao mai ke kikowaena a i ke kikowaena: i hōʻike ʻia \(c\), me ka pilina elliptical maʻamau:
\[
c^2 = a^2 – b^2
\]
Hoʻomaka pinepine kahi paio o nā manaʻo ma ʻaneʻi: i loko o kahi ellipse, paʻa mau ʻo \(a \ge b\) a aia nā foci ma ke axis nui.
Eia kekahi, aia ke kumumanaʻo o ka eccentricity \(e\) e ana ana i ka "slant waho" o kahi ellipse:
\[
e = \frac{c}{a}, \quad 0 \le e < 1
\]
Jika \(e = 0\), elips menjadi lingkaran (karena \(c = 0\), fokus berimpit di pusat).
2. Persamaan Standar Elips Berpusat di Titik Asal
Jika elips berpusat di titik asal \((0,0)\) dan sumbu-sumbunya sejajar sumbu koordinat, persamaan elips memiliki bentuk standar yang sangat dikenal.
a) Sumbu mayor horizontal
Jika sumbu mayor sejajar sumbu-\(x\), maka:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
dengan \(a > b\). Fokus terletak pada sumbu-\(x\), yaitu di titik:
\[
(\pm c, 0), \quad \text{me } c^2 = a^2 – b^2
\]
b) ʻO ke koʻi nui kū pololei
Inā kūlike ke axis nui me ke axis-y, a laila:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
me \(a > b\). Aia ke kiko ma ka axis \(y\), ʻo ia hoʻi:
\[
(0, \pm c), \quad c^2 = a^2 – b^2
\]
ʻO kēia ʻano maʻamau e maʻalahi ai ka heluhelu ʻana i nā ʻano o kahi ellipse: ʻo nā waiwai o \(a\) a me \(b\) e hōʻike pololei ana i ka nui o ka ellipse, ʻoiai ʻo \(c\) e hoʻoholo ana i ke kūlana o nā foci.
3. ʻO ke kaulike Ellipse i kikowaena ʻia ma \((h,k)\)
I loko o nā pilikia geometry analytical he nui, ʻaʻole i waenakonu mau ka ellipse ma ke kikowaena hoʻonohonoho. Inā waenakonu ka ellipse ma \((h,k)\), a laila loli ka hoohalike maʻamau i:
a) ʻO ke koʻi nui ākea
\[
\frac{(xh)^2}{a^2} + \frac{(yk)^2}{b^2} = 1
\]
b) ʻO ke koʻi nui kū pololei
\[
\frac{(xh)^2}{b^2} + \frac{(yk)^2}{a^2} = 1
\]
ʻO kēia hoʻololi he neʻe wale nō ia (unuhi) o ka ellipse, kahi i kikowaena mua ʻia ma ke kumu. Ke neʻe nei hoʻi ke kiko i ke kikowaena hou:
– No ke axis nui ʻākea: \((h \pm c, k)\)
– No ke axis nui kū pololei: \((h, k \pm c)\)
4. Mai ka Wehewehena o ke Kūleʻa a i ka Hoʻohālikelike o kahi Ellipse
Hiki ke hoʻohana ʻia ka wehewehe ʻana o kahi ellipse ma ke ʻano he huina o nā mamao i ʻelua mau foci mau ma ke ʻano he kumu no ka loaʻa ʻana o nā kaulike. No ka laʻana, e manaʻo ʻia aia nā foci ma \((c,0)\) a me \((-c,0)\), a ʻo kahi kiko ma ka ellipse ʻo \((x,y)\). ʻO nā mamao o kēlā kiko i kēlā me kēia kiko:
\[
d_1 = \sqrt{(xc)^2 + y^2}, \quad d_2 = \sqrt{(x+c)^2 + y^2}
\]
No ka mea, ua mau ka nui:
\[
d_1 + d_2 = 2a
\]
Ma ka hoʻoponopono algebraic (e hoʻopaʻa ana i ʻelua manawa e hoʻopau i nā aʻa), loaʻa iā mākou ka hoohalike:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
me \(b^2 = a^2 – c^2\). Hōʻike kēia ʻaʻole wale ke ʻano maʻamau o ka ellipse he ʻano "hoʻopaʻanaʻau", akā ua hele maoli mai kahi wehewehe geometric.
5. Ka Hoʻohālikelike Laulā o kahi Ellipse a me kona ʻIke ʻana
I ka hana maoli, loaʻa pinepine mākou i nā hoʻohālikelike quadratic me ʻelua mau loli ʻaʻole i ke ʻano maʻamau, no ka laʻana:
\[
ʻAx^2 + By^2 + Cx + Dy + E = 0
\]
Hiki i kahi hoʻohālikelike e like me kēia ke hōʻike i kahi ellipse, kahi parabola, a i ʻole kahi hyperbola. No ka hōʻoia ʻana he ellipse ia (me nā axis e kūlike ana i nā coordinates), ma ke ʻano maʻamau ʻo \(A\) a me \(B\) pono:
- hōʻailona like (maikaʻi a maikaʻi ʻole paha),
– a ʻaʻole like ka nui (inā like ko lākou nui a ʻaʻohe huaʻōlelo \(xy\), he mea hiki loa ke ʻano he pōʻai).
No ka hoʻololi ʻana i ke ʻano ellipse maʻamau, ʻo ke ʻano hana i hoʻohana pinepine ʻia ʻo ia ka hoʻopau ʻana i ka huinahā ma nā huaʻōlelo \(x\) a me \(y\). He laʻana maʻalahi:
\[
4x^2 + 9y^2 – 8x + 18y – 5 = 0
\]
Pūʻulu:
\[
4(x^2 – 2x) + 9(y^2 + 2y) = 5
\]
E hoʻopiha i ka huinahā:
\[
4[(x-1)^2 – 1] + 9[(y+1)^2 – 1] = 5
\]
\[
4(x-1)^2 + 9(y+1)^2 = 5 + 4 + 9 = 18
\]
No 18:
\[
\frac{(x-1)^2}{\frac{18}{4}} + \frac{(y+1)^2}{2} = 1
\]
ʻo ia ke ʻano maʻamau o kahi ellipse me ke kikowaena \((1,-1)\).
6. Nā Hoʻohana o Ellipses i ke Geometry a me ke Ola Maoli
ʻAʻole nā mea kumumanaʻo wale nō nā ellipses. I ke geometry a me ka ʻepekema i hoʻopili ʻia, he kuleana nui nā ellipses:
1. Astronomy (Ke Kānāwai o Kepler): he elliptical ke kaapuni o kahi honua me ka Lā ma kekahi kiko.
2. Optics a me acoustics: ʻo ka waiwai o ka elliptical reflection e ʻōlelo ana e hōʻike ʻia nā nalu mai kekahi kikowaena ma o kekahi kikowaena ʻē aʻe. Hoʻohana ʻia kēia i ka hoʻolālā ʻana o nā hale ʻaha mele a i ʻole kekahi mau aniani reflector.
3. ʻEnekinia mīkini: hoʻohana kekahi mau mīkini kia a i ʻole nā mīkini cam i nā ala elliptical.
4. Kuhikuhipuʻuone: hāʻawi ke ʻano elliptical i ka hui pū ʻana o nā aesthetics a me ka hana acoustic.
Ma ka hoʻomaopopo ʻana i ka hoohalike ellipse, hiki iā mākou ke kālailai i ka nui, ke kūlana, a me nā waiwai o nā trajectories i nā ʻōnaehana like ʻole.
7. Manaʻo
ʻO ke kaulike o kahi ellipse i ke geometry e hoʻopili i ka hakahaka ma waena o ka wehewehe geometric (ka huina o nā mamao i ʻelua mau foci mau) a me ka hōʻike analytical (kahi kaulike algebraic i nā coordinates). ʻO ke ʻano maʻamau o kahi ellipse e maʻalahi ai ka ʻike ʻana i ke kikowaena, nā lōʻihi o nā axes, a me nā kūlana o nā foci, ʻoiai hiki ke hoʻololi ʻia nā ʻano laulā i ke ʻano maʻamau ma ka hoʻopau ʻana i ka square. ʻO ka hoʻomaopopo ʻana i nā ellipses ʻaʻole wale e kōkua i ka hoʻoponopono ʻana i nā pilikia geometry analytical akā wehe pū i ka ʻike i ke ʻano o ka wehewehe ʻana o ka makemakika i nā hanana kūlohelohe e like me nā orbits planetary a me nā waiwai o ka noʻonoʻo nalu.
Inā makemake ʻoe, hiki iaʻu ke hoʻohui i nā pilikia hoʻohālike a hoʻopau i nā kūkākūkā (e.g. ke hoʻoholo ʻana i ka nānā ʻana, ka eccentricity, a i ʻole ke kaha kiʻi ʻana i kahi sketch o kahi ellipse mai kona hoohalike).
