Tn̂g-ko-îⁿ

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 e {\displaystyle e} (Hàn-jī: 離心率; Eng-gí: eccentricity), chit-ê siàu ê hoān-ûi sī tùi e = 0 {\displaystyle e=0} (oân-choân bô píⁿ, tiō-sī îⁿ-hêng, che sī chi̍t ê chì-ke̍k chōng-hóng) it-ti̍t kàu e = 1 {\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.
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.
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
Tùi jīm-hô chi̍t ê tiám ( x , y ) {\displaystyle (x,y)} , i kàu kî-tiong chi̍t ê chiau-tiám ( c , 0 ) {\displaystyle (c,0)} ê kī-lī is ( x − c ) 2 + y 2 {\textstyle {\sqrt {(x-c)^{2}+y^{2}}}} , kàu lēng-gōa chi̍t ê chiau-tiám ( − c , 0 ) {\displaystyle (-c,0)} ê kī-lī is ( x + c ) 2 + y 2 {\textstyle {\sqrt {(x+c)^{2}+y^{2}}}} . Só͘-í kiàn-pái nā-sī ū chi̍t ê tiám ( x , y ) {\displaystyle (x,y)} hû-ha̍p
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 {\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:
Lán mā sī ē-sái khì chhōe y ê kái, tiō-sī:
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 ) {\displaystyle (X,\,Y)} ê boán-chiok chit ê bô-bêng-hián ê hong-têng-sek:
kî-tiong it-tēng ài ū B 2 − 4 A C < 0. {\displaystyle B^{2}-4AC<0.}
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 θ {\displaystyle \theta } , án-ne tn̂g-ko-îⁿ ê hong-têng-sek tiō-sī
nā-sī kā tn̂g-ko-îⁿ ê tiong-ng-tiám kǹg tī goân-tiám, kak chō-piau θ {\displaystyle \theta } oân-á sī tùi tn̂g-kèng khai-sí niû, án-ne tn̂g-ko-îⁿ ê hong-têng-sek tiō-sī
Koh khah it-poaⁿ ê chêng-hêng sī
Chia-ê kak-tō͘ θ {\displaystyle \theta } hō-chò chin-kīn-tiám-kak. Koh-ū, chia-ê kong-sek ê hun-chú lóng sī pòaⁿ-chiàⁿ-chiau-hiân ℓ = a ( 1 − e 2 ) {\displaystyle \ell =a(1-e^{2})} .
Tn̂g-ko-îⁿ ē-sái ēng saⁿ-kak koan-sò͘ lâi piáu-sī:
kî-tiong ê t {\textstyle t} (thian-bûn-ha̍k siōng kiò-chò chin-kīn-tiám-kak) m̄ sī ( x ( t ) , y ( t ) ) {\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.
Hō͘ chi̍t-ê tn̂g-ko-îⁿ pau ·khí-lâi ê Bīn-chek sī
kî-tiong a {\textstyle a} kap b {\textstyle b} hun-pia̍t sī tn̂g-ko-îⁿ ê pòaⁿ-tn̂g-kèng kap pòaⁿ-té-kèng. Che ē-sái án-ne lâi chèng-bêng:
Tē-jī ê chek-hun tiō-sī chi̍t ê îⁿ-hêng ê bīn-chek, i ê pòaⁿ-kèng sī a {\textstyle a} , bīn-chek tiō-sī π a 2 {\textstyle \pi a^{2}} .
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.