Wāhanga Kōnika Pūngāwerewere
Pendahuluan
I roto i te pāngarau, inā koa i roto i te āhuahanga tātari, he kaupapa whakamere, he kaupapa whānui hoki ngā wāhanga kōniko. E whā ngā momo matua o ngā wāhanga kōniko: ngā porowhita, ngā porowhita, ngā parabola, me ngā hyperbola. I roto i tēnei tuhinga, ka arotahi mātou ki tētahi o ēnei momo: te hyperbola. He āhua me ngā āhuatanga ahurei ngā hyperbola ki te whakatauritea ki ētahi atu wāhanga kōniko, ā, he whānui te whakamahinga i roto i ngā momo mara, tae atu ki te whetū arorangi, te ahupūngao, me te hangarau.
Ngā Whakamāramatanga me ngā Ariā Taketake
Ko te hyperbola he huinga pūwāhi i roto i te papa e pumau ana te uara tino o te rerekētanga o ō rātou tawhiti mai i ngā pūwāhi pumau e rua e kiia nei ko ngā arotahi. Mēnā he pūwāhi pumau e rua a F₁ me F₂ i roto i te papa, ko te hyperbola te huinga o ngā pūwāhi katoa P(x, y) kia rite ki te |d(P, F₁) – d(P, F₂)| = k, ko k he pūwāhi pai, ā, he iti iho i te tawhiti i waenganui i a F₁ me F₂.
I te nuinga o te wā, mō ngā arotahi F₁(c, 0) me F₂(-c, 0), ka taea te tuhi i te āhua paerewa o te whārite hyperbola kei te pūtake (0,0) tōna pokapū penei:
\[ \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 \]
ranei
\[ \frac{y^2}{a^2} – \frac{x^2}{b^2} = 1 \]
ko a² + b² = c².
He tikanga āhuahanga ō ngā tawhā a, b, me c i roto i te horopaki o te hyperbola:
– a : Te tawhiti mai i te pokapū ki ia pito i runga i te tuaka matua.
– b : Te tawhiti mai i te pokapū ki te pūwāhi i runga i te tuaka iti e whiti ana i te tuaka matua.
– c : Te tawhiti mai i te pokapū ki ia arotahi.
Ngā Āhuatanga Āhua-Pūrua
Ko tētahi o ngā āhuatanga matua o te hyperbola ko te noho mai o ngā asymptotes. Ko ngā asymptotes ngā rārangi e neke ai te hyperbola i a ia e whakatata atu ana ki te mutunga kore. Ka tohu ēnei i te ahunga e neke atu ai te hyperbola i tōna pokapū. Mō te hyperbola o te āhua paerewa \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 \), ka hoatuhia ngā asymptotes e te whārite:
\[ y = \pm \frac{b}{a} x \]
Ka taea te whakaaro ki ngā asymptotes he "ārahi" e whakaatu ana i te horapa o ngā peka o te hyperbola ki waho.
Ngā Āhua me te Whakarōpūtanga o te Hyperbola
Ka taea te whakarōpū i ngā hyperbolas i runga i tō rātou aronga:
1. Pūngao Whakapae: Mena ko te āhua paerewa ko \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 \), ka tuwhera te pūngao ki te taha matau me te taha maui. He ōrite te āhua o ōna manga ki te tuaka-x.
2. Pūngao Whānui Poutū: Mena ko te āhua paerewa ko \(\frac{y^2}{a^2} – \frac{x^2}{b^2} = 1 \), ka puare ake, ka puare iho te pūngao whānui. He ōrite te āhua o ōna manga huri noa i te tuaka-y.
Te rerekētanga o te Whakanuia
Ko te āhua rerekē, e tohuhia ana e te e, he tawhā e ine ana i te kaha o te "piko" o te hyperbola. Ko te āhua rerekē mō te hyperbola ka hoatu e te tātai:
\[ e = \frac{c}{a} \]
I te mea he nui ake a c i te a mō tētahi hyperbola, he nui ake te rerekētanga o tētahi hyperbola i te 1 (e > 1). Ka nui ake te rerekētanga, ka papatahi ake, ka roa ake hoki te hyperbola.
Ahupūngao me ngā Whakamahinga o te Whakapūtanga
Ehara i te mea he mea nui ngā hyperbola i roto i te ao o te ariā pāngarau anake, engari i roto hoki i ngā tono mahi maha:
1. Te Tātaritanga whetū:
– Ka puta ngā hyperbola i roto i ngā porowhita hyperbic o ngā wheturangi me ētahi atu tinana rangi e toro mai ana ki tō tātou pūnaha rā, engari he tere rawa ō rātou ara hei mawhiti i te kukume ā-papa o te rā.
2. Ngā Tirohanga me te Whakaata:
– I roto i te hangarau whatu, ka whakamahia ngā whakaata pūrua hei arotahi i te mārama. Kāore i rite ki ngā whakaata parabolic, ka taea e ngā whakaata pūrua te hopu i te mārama mai i ngā pūwāhi arotahi e rua.
3. Whakatere me te Tauwāhi:
– I roto i ngā pūnaha whakatere (pērā i te LORAN me ngā pūnaha tūnga hoa-hoa-ranei-hoariri (IFF)), ko te kaupapa matua o te mahi e ahu mai ana i te ine i te rerekētanga o ngā wā taenga mai o ngā tohu rerekē e rua e whakaputa ana i tētahi kōpiko pūwero i runga i te whenua.
4. Te Hikohiko me te Whakawhitiwhiti Kōrero:
– E whakamahia ana ngā hyperbola mō te hoahoa pūihi me te whakatauira i te tohatoha pūngao i roto i ngā wāhanga hiko, ā, kua whakamātauhia he tino pai mō ngā tono whakawhitiwhiti kōrero.
Whakamutunga
Ko te hyperbola, he momo wāhanga kōnika, he maha ngā āhuatanga pāngarau me ngā tono mahi nui. Mā te mārama ki tōna whakamāramatanga, ngā whārite paerewa, ngā tawhā nui pēnei i te a, b, me te c, me te mārama ki tōna āhua rerekē me ngā asymptotes, ka taea e tātou te ruku hohonu ki ngā tono o te ao tūturu o tēnei āhua ā-ira i roto i te pūtaiao me te hangarau. E whakaatu ana te hyperbola i te ataahua me te uaua o te pāngarau i roto i te whakatauira i ngā āhuatanga taiao me ngā hangarau hou. Mā te mārama ki ōna ariā me ngā tono taketake, kāore e taea e tātou te maioha ki ōna āhua pāngarau anake engari ka taea hoki te whakamahi hei whakaoti rapanga o te ao tūturu.
