Tn̂g-ko-îⁿ

Sò͘-ha̍k siōng, **tn̂g-ko-îⁿ** (; tn̂g-tu-îⁿ; ) sī chi̍t khoán pêⁿ-bīn khiok-sòaⁿ, i ûi-tio̍h nn̄g-ê chiau-tiám (; ), 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 e (; ), chit-ê siàu ê hoān-ûi sī tùi e=0 (oân-choân bô píⁿ, tiō-sī îⁿ-hêng, che sī chi̍t ê chì-ke̍k chōng-hóng) it-ti̍t kàu 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 F_1=(c,0) kap  F_2=(-c,0),
: tèng-tiám sī V_1=(a,0) kap V_2=(-a,0).

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

: \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 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:

: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1.

Lán mā sī ē-sái khì chhōe *y* ê kái, tiō-sī:

: 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 (X,\, Y) ê boán-chiok chit ê bô-bêng-hián ê hong-têng-sek:

: AX^2 + B X Y + C Y^2 + D X + E Y + F = 0

kî-tiong it-tēng ài ū B^2 - 4AC 
### Chham-sò͘

* Tn̂g-kó-îⁿ ê **chú-iàu-kèng** (; ) 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** (; ), i ê chi̍t-pòaⁿ kiò-chò **pòaⁿ-tn̂g-kèng** (; ), tn̂g-tō͘ kì-hō sī a;
** khah té ·ê kiò-chò **té-kèng** (; ), i ê chi̍t-pòaⁿ **pòaⁿ-té-kèng** (; ), tn̂g-tō͘ kì-hō sī b. 
** Lán tio̍h iau-kiû a\geq b>0. 
* **Sòaⁿ-sèng lī-sim-lu̍t** (; ) sī chiau-tiám kàu tiong-ng ê kī-lī: c=\sqrt{a^2-b^2}.
* **Lī-sim-lu̍t** sī c kap a ê pí-lē: e=c/a=\sqrt{1-\left(b/a\right)^2}.
* **Chiàⁿ-chiau-hiân** (; ) 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** (; ). I ē-sái án-ne lâi kè-sǹg: \ell=b^2/a = a\left(1 - e^2\right).縮圖|301x301像素|Tn̂g-ko-îⁿ ê chham-sò͘:  * ''<math>a</math>'': pòaⁿ-tn̂g-kèng, * ''<math>b</math>'': pòaⁿ-té-kèng, * ''<math>c</math>'': sòaⁿ-sèng lī-sim-lu̍t, * ''<math>p</math>'': pòaⁿ-chiàⁿ-chiau-hiân (it-poaⁿ sī siá <math>\ell</math>).

## Tī kek chō-piau-hē

### Tùi tiong-ng-tiám

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 \theta, án-ne tn̂g-ko-îⁿ ê hong-têng-sek tiō-sī
: r(\theta) = \frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}=\frac{b}{\sqrt{1 - (e\cos\theta)^2}}.

### Tùi chiau-tiám

nā-sī kā tn̂g-ko-îⁿ ê tiong-ng-tiám kǹg tī goân-tiám, kak chō-piau \theta oân-á sī tùi tn̂g-kèng khai-sí niû, án-ne tn̂g-ko-îⁿ ê hong-têng-sek tiō-sī
: r(\theta)=\frac{a (1-e^2)}{1 \pm e\cos\theta}.

Koh khah it-poaⁿ ê chêng-hêng sī 

:r(\theta)=\frac{a (1-e^2)}{1 - e\cos(\theta - \phi)}.

Chia-ê  kak-tō͘ \theta hō-chò chin-kīn-tiám-kak. Koh-ū, chia-ê kong-sek ê hun-chú lóng sī pòaⁿ-chiàⁿ-chiau-hiân \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:
: (x,y)=(a\cos{t},b\sin{t}),\ 0\leq t 
kî-tiong ê t (thian-bûn-ha̍k siōng kiò-chò chin-kīn-tiám-kak) m̄ sī (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.
縮圖

## Sèng-chit

### Bīn-chek
Hō͘ chi̍t-ê tn̂g-ko-îⁿ pau ·khí-lâi ê Bīn-chek sī
: A=\pi ab,
kî-tiong a kap b hun-pia̍t sī tn̂g-ko-îⁿ ê pòaⁿ-tn̂g-kèng kap pòaⁿ-té-kèng. Che ē-sái án-ne lâi chèng-bêng:
: \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 a, bīn-chek tiō-sī \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.

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

無|縮圖|de La Hire hong-hoat ê tōng-ōe

### Hn̂g-teng hoat

# Kā nn̄g ê tōa-thâu-teng tèng tī chóa bīn-téng, chiâⁿ-chò tn̂g-ko-îⁿ ê chiau-tiám. 
# Kā chi̍t tiâu sòaⁿ ê siang-thâu pa̍k tī tōa-thâu-teng, sòaⁿ ê tn̂g-tō͘ sī 2a.
# 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.

無|縮圖|Hn̂g-teng hoat

## Chham-khó chu-liāu


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