Sò͘-ha̍k siōng, **tn̂g-ko-îⁿ** (Hàn-jī: 長篙圓; tn̂g-tu-îⁿ; Eng-gí: ellipse) sī chi̍t khoán pêⁿ-bīn khiok-sòaⁿ, i ûi-tio̍h nn̄g-ê chiau-tiám (Hàn-jī: 焦點; Eng-gí: focus), tì-sú só͘-ū tī khiok-sòaⁿ téng-koân ê tiám, kàu nn̄g-ê chiau-tiám kī-lī ê chóng-hô lóng kò͘-tēng. Tn̂g-ko-îⁿ khiā-sǹg-kóng sī chi̍t chióng píⁿ-píⁿ ê îⁿ-hêng, a îⁿ-hêng khiā-sǹg-kóng sī chi̍t chióng te̍k-sû ê tn̂g-ko-îⁿ, in-ūi îⁿ-hêng ê nn̄g-ê chiau-tiám sī tha̍h chò-hóe ·ê. Kóng chi̍t ê tn̂g-ko-îⁿ ū góa píⁿ ê thêng-tō͘ kiò-chò lī-sim-lu̍t {\displaystyle e} (Hàn-jī: 離心率; Eng-gí: eccentricity), chit-ê siàu ê hoān-ûi sī tùi {\displaystyle e=0} (oân-choân bô píⁿ, tiō-sī îⁿ-hêng, che sī chi̍t ê chì-ke̍k chōng-hóng) it-ti̍t kàu {\displaystyle e=1} (píⁿ kàu chīn-pōng, che mā sī chi̍t ê chì-ke̍k chōng-hóng; m̄-koh che i í-keng m̄ sī tn̂g-ko-îⁿ, soah chiâⁿ-chò khian-sòaⁿ ·ah.)

Tn̂g-ko-îⁿ tī bu̍t-lí-ha̍k, thian-bûn-ha̍k, kap kang-têng-ha̍k lóng chin chiap ū lō͘-ēng. Phì-lūn-kóng, tī lán Thài-iông-hē lāi-té, múi chi̍t lia̍p he̍k-chhiⁿ ê kúi-tō lóng chha-put-to sī tn̂g-ko-îⁿ, jî-chhiáⁿ ji̍t-thâu tiō-sī tiám tī kî-tiong chi̍t ê chiau-tiám (kóng khah cheng-khak ·leh, ji̍t-thâu kap he̍k-chhiⁿ ha̍p chò chi̍t cho͘ hē-thóng ê sî, kúi-tō ê chiau-tiám tiō-sī tī-teh in tiōng-sim ê só͘-chāi. Che mā sī Kepler tē-it tēng-lu̍t teh kóng ·ê.

Miâ-chheng
----------

Tn̂g-ko-îⁿ ê Eng-gí ellipse sī tùi Hi-lia̍p-gí ἔλλειψις (*élleipsis*, "so͘-hut") lâi ·ê, che sī kó͘-chá Hi-lia̍p sò͘-ha̍k-ka Apollonius tī i ê chheh *Conics* lāi-té hoat-bêng ·ê.

Tī Tâi-gí, tn̂g-ko-îⁿ ū chē-chē chióng kóng-hoat, pau-koat Hàn-jī-sû **thó-îⁿ** (橢圓), koh ū ah-nn̄g-hêng (鴨卵形), io-chí-hoān (腰子範), ke-nn̄g-hêng (雞卵形), ti-io-hêng (豬腰形), ti-io-hoān (豬腰範), kap tn̂g-tu-îⁿ (長株圓), chhó-kan-ná-hêng (草橄欖形) téng-téng. Chóng-·sī chia ê sû-lūi iáu bô phiau-chún ê kóng-hoat.

Tī khut-chhioh-kak chō-piau-hē
------------------------------

### Phiau-chún hong-têng-sek

Tn̂g-ko-îⁿ tī khut-chhioh-kak chō-piau-hē ê phiau-chún hêng-sek sī kā tn̂g-ko-îⁿ ê tiong-ng kǹg tī goân-tiám, the *x te̍k* sī chú-iàu-te̍k, jî-chhiáⁿ

chiau-tiám sī {\textstyle F\_{1}=(c,0)} kap {\textstyle F\_{2}=(-c,0)},
tèng-tiám sī {\textstyle V\_{1}=(a,0)} kap {\textstyle V\_{2}=(-a,0)}.
Tùi jīm-hô chi̍t ê tiám {\displaystyle (x,y)}, i kàu kî-tiong chi̍t ê chiau-tiám {\displaystyle (c,0)} ê kī-lī is {\textstyle {\sqrt {(x-c)^{2}+y^{2}}}}, kàu lēng-gōa chi̍t ê chiau-tiám {\displaystyle (-c,0)} ê kī-lī is {\textstyle {\sqrt {(x+c)^{2}+y^{2}}}}. Só͘-í kiàn-pái nā-sī ū chi̍t ê tiám {\displaystyle (x,y)} hû-ha̍p 

{\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}+{\sqrt {(x+c)^{2}+y^{2}}}=2a}
ê sî, chit ê tiám tiō tiám tī tn̂g-ko-îⁿ ê téng-koân. Lán nā kā téng-koân hit chi̍t sek chò chi̍t-kóa sek-tòng ê pêng-hong, kā kin-hō the̍h-tiāu, koh sú-iōng {\textstyle b^{2}=a^{2}-c^{2}}  (kòaⁿ piⁿ-·á ê tô͘), kàu-bóe tiō ē tàu-chhut tn̂g-ko-îⁿ ê phiau-chún hong-têng-sek:

{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}
Lán mā sī ē-sái khì chhōe *y* ê kái, tiō-sī:

{\displaystyle y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}=\pm {\sqrt {\left(a^{2}-x^{2}\right)\left(1-e^{2}\right)}}.}
### It-poaⁿ hêng-sek

Kái-sek kí-hô-ha̍k tiong, tn̂g-ko-îⁿ sī chi̍t khoân jî-chhú hêng-sek. Tī hui-thòe-hòa ê chōng-hóng chi-hā, khut-chhioh-kak chō-piau pêⁿ-bīn téng-koân ê tiám {\displaystyle (X,\,Y)} ê boán-chiok chit ê bô-bêng-hián ê hong-têng-sek:

{\displaystyle AX^{2}+BXY+CY^{2}+DX+EY+F=0}
kî-tiong it-tēng ài ū {\displaystyle B^{2}-4AC<0.}

### Chham-sò͘

* Tn̂g-kó-îⁿ ê **chú-iàu-kèng** (Hàn-jī: 主要徑; Eng-gí: principle axes) ū nn̄g tiâu, in hō͘-siong sûi-ti̍t, mā lóng sī tn̂g-ko-îⁿ ê tùi-thīn-te̍k.
	+ Khah tn̂g ·ê kiò-chò **tn̂g-kèng** (Hàn-jī: 長徑; Eng-gí: major axis), i ê chi̍t-pòaⁿ kiò-chò **pòaⁿ-tn̂g-kèng** (Hàn-jī: 半長徑; Eng-gí: semi-major axis), tn̂g-tō͘ kì-hō sī {\textstyle a};
	+ khah té ·ê kiò-chò **té-kèng** (Hàn-jī: 短徑; Eng-gí: minor axis), i ê chi̍t-pòaⁿ **pòaⁿ-té-kèng** (Hàn-jī: 半短徑; Eng-gí: semi-minor axis), tn̂g-tō͘ kì-hō sī {\textstyle b}.
	+ Lán tio̍h iau-kiû {\textstyle a\geq b>0}.
* **Sòaⁿ-sèng lī-sim-lu̍t** (Hàn-jī: 線性離心率; Eng-gí: linear eccentricity) sī chiau-tiám kàu tiong-ng ê kī-lī: {\textstyle c={\sqrt {a^{2}-b^{2}}}}.
* **Lī-sim-lu̍t** sī {\textstyle c} kap {\textstyle a} ê pí-lē: {\textstyle e=c/a={\sqrt {1-\left(b/a\right)^{2}}}}.
* **Chiàⁿ-chiau-hiân** (Hàn-jī: 正焦弦; Eng-gí: latus rectum) tiō-sī chhǹg-kòe chi̍t ê chiau-tiám ê chi̍t tiâu hiân, koh kah tn̂g-kèng sûi-ti̍t. I ê chi̍t-pòaⁿ sī kiò-chò **pòaⁿ-chiàⁿ-chiau-hiân** (Hàn-jī: 半正焦弦; Eng-gí: semi-latus rectum). I ē-sái án-ne lâi kè-sǹg: {\textstyle \ell =b^{2}/a=a\left(1-e^{2}\right)}.Tn̂g-ko-îⁿ ê chham-sò͘: \* *{\displaystyle a}*: pòaⁿ-tn̂g-kèng, \* *{\displaystyle b}*: pòaⁿ-té-kèng, \* *{\displaystyle c}*: sòaⁿ-sèng lī-sim-lu̍t, \* *{\displaystyle p}*: pòaⁿ-chiàⁿ-chiau-hiân (it-poaⁿ sī siá {\displaystyle \ell }).

Tī kek chō-piau-hē
------------------

### Tùi tiong-ng-tiám

Polar coordinates centered at the center.
Tī kek chō-piau-hē lāi-té, nā-sī kā tn̂g-ko-îⁿ ê tiong-ng-tiám kǹg tī goân-tiám, pēng-chhiáⁿ tùi tn̂g-kèng khai-sí niû kak chō-piau {\displaystyle \theta }, án-ne tn̂g-ko-îⁿ ê hong-têng-sek tiō-sī

{\displaystyle r(\theta )={\frac {ab}{\sqrt {(b\cos \theta )^{2}+(a\sin \theta )^{2}}}}={\frac {b}{\sqrt {1-(e\cos \theta )^{2}}}}}.
### Tùi chiau-tiám

Polar coordinates centered at focus.
nā-sī kā tn̂g-ko-îⁿ ê tiong-ng-tiám kǹg tī goân-tiám, kak chō-piau {\displaystyle \theta } oân-á sī tùi tn̂g-kèng khai-sí niû, án-ne tn̂g-ko-îⁿ ê hong-têng-sek tiō-sī

{\displaystyle r(\theta )={\frac {a(1-e^{2})}{1\pm e\cos \theta }}}.
Koh khah it-poaⁿ ê chêng-hêng sī 

{\displaystyle r(\theta )={\frac {a(1-e^{2})}{1-e\cos(\theta -\phi )}}.}
Chia-ê kak-tō͘ {\displaystyle \theta } hō-chò chin-kīn-tiám-kak. Koh-ū, chia-ê kong-sek ê hun-chú lóng sī pòaⁿ-chiàⁿ-chiau-hiân {\displaystyle \ell =a(1-e^{2})}.

Chham-sò͘ piáu-sī
-----------------

### Phiau-chún chham-sò͘ piáu-sī

Tn̂g-ko-îⁿ ē-sái ēng saⁿ-kak koan-sò͘ lâi piáu-sī:

{\displaystyle (x,y)=(a\cos {t},b\sin {t}),\ 0\leq t<2\pi .}
kî-tiong ê {\textstyle t} (thian-bûn-ha̍k siōng kiò-chò chin-kīn-tiám-kak) m̄ sī {\displaystyle (x(t),y(t))} kap *x*-te̍k chi-kan ê kak-tō͘, i ê kí-hô-ha̍k ì-gī sī kap de La Hire hong-hoat ū koan-hē.

Sèng-chit
---------

### Bīn-chek

Hō͘ chi̍t-ê tn̂g-ko-îⁿ pau ·khí-lâi ê Bīn-chek sī

{\displaystyle A=\pi ab,}
kî-tiong {\textstyle a} kap {\textstyle b} hun-pia̍t sī tn̂g-ko-îⁿ ê pòaⁿ-tn̂g-kèng kap pòaⁿ-té-kèng. Che ē-sái án-ne lâi chèng-bêng:

{\displaystyle {\begin{aligned}A&=\int \_{-a}^{a}2b{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\,\mathrm {d} x\\&={\frac {b}{a}}\int \_{-a}^{a}2{\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x.\end{aligned}}}
Tē-jī ê chek-hun tiō-sī chi̍t ê îⁿ-hêng ê bīn-chek, i ê pòaⁿ-kèng sī {\textstyle a}, bīn-chek tiō-sī {\textstyle \pi a^{2}}.

Ōe tn̂g-ko-îⁿ ê hoat-tō͘
------------------------

### de La Hire hoat-tō͘

Chit ê hoat-tō͘ ê ki-chhó͘ sī tn̂g-ko-îⁿ ê phiau-chún chham-sò͘ piáu-sī. I sī Philippe de La Hire hoat-bêng ·ê.

1. Ōe nn̄g-ê *îⁿ-hêng*, hō͘ in tiong-ng-tiám sio-kâng (mā ē chiâⁿ-chò tn̂g-ko-îⁿ ê tiong-ng-tiám), pòaⁿ-kèng hun-pia̍t sī {\textstyle a} kap {\textstyle b}.
2. Ōe *chi̍t tiâu sòaⁿ chhǹg-kòe tiong-ng-tiám*, hō͘ i kah nn̄g-ê îⁿ-hêng hun-pia̍t kau-chhe tī tiám {\textstyle A} kap tiám {\textstyle B}.
3. Ōe *chi̍t tiâu sòaⁿ* *chhǹg-kòe* {\textstyle A}, hō͘ i kah té-kèng pêng-hêng, koh ōe *chi̍t tiâu sòaⁿ* *chhǹg-kòe* {\textstyle B}, hō͘ i kah tn̂g-kèng pêng-hêng Chit nn̄g tiâu sòaⁿ ē kau-chhe tī tn̂g-ko-îⁿ ê téng-koân.
4. Sú-iōng bô-kâng ê sòaⁿ chhng-kòe tiong-sim-tiám, tiông-ho̍k chò tē (2) pō͘ kap tē (3) pō͘.

de La Hire hong-hoat ê tōng-ōe
### Hn̂g-teng hoat

1. Kā nn̄g ê tōa-thâu-teng tèng tī chóa bīn-téng, chiâⁿ-chò tn̂g-ko-îⁿ ê chiau-tiám.
2. Kā chi̍t tiâu sòaⁿ ê siang-thâu pa̍k tī tōa-thâu-teng, sòaⁿ ê tn̂g-tō͘ sī {\displaystyle 2a}.
3. Ēng pit-bóe kā sòaⁿ peⁿ hō͘ ân ē sî, ōe-chhut chi̍t kho͘ khiok-sòaⁿ, chit-ê khiok-sòaⁿ tiō sī tn̂g-ko-îⁿ.

Hn̂g-teng hoat

Chham-khó chu-liāu
------------------

1. ↑  Embree, B. (1973). *Tāi-ing sî-tén.* Tē-286 ia̍h.
2. ↑  台華線頂對照典
3. ↑  Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006). "Chapter 10". *Precalculus with Limits*. Cengage Learning. p. 767. ISBN 978-0-618-66089-6.
4. ↑  Young, Cynthia Y. (2010). "Chapter 9". *Precalculus*. John Wiley and Sons. p. 831. ISBN 978-0-471-75684-2.