Āhuahanga Tātari: Te Hononga i te Āraipa me te Āhuahanga
Ko te āhuahanga tātari he peka o te pāngarau e whakakotahi ana i ngā ariā taurangi me te āhuahanga hei ako i ngā whanaungatanga me ngā āhuatanga āhuahanga mā te whakamahi i ngā whārite taurangi. I whakawhanakehia motuhaketia e René Descartes rāua ko Pierre de Fermat i te rautau 17, ko te āhuahanga tātari te pūtake mō ngā ariā maha i roto i te pāngarau hou, te ahupūngao, me te hangarau. Ka tūhuratia e tēnei tuhinga ngā ariā taketake, ngā tono, me te hiranga o te āhuahanga tātari i roto i ngā momo mara pūtaiao.
Hītori me te Whanaketanga
He roa te hītori o te whanaketanga motuhake o te āhuahanga me te arapū i mua i te hanumitanga ki te āhuahanga tātari. Ko ngā ine āhuahanga Kariki onamata pērā i a Euclid rāua ko Archimedes i mahi noa me ngā ariā āhuahanga, kāore i te whakamahi i ngā whārite. I tētahi atu taha, i whanake te Arapū i roto i te ao Ihirama i te rautau 9 me te 10 mā roto i ngā mahi a ngā tohunga pāngarau pērā i a Al-Khwarizmi.
I te rautau 17, i hurihia te pāngarau e te mahi a Descartes rāua ko Fermat, nā rāua i whakauru mai he tikanga mō te whakamāori i ngā raruraru ā-ira ki te āhua tau mā te whakamahi i ngā taunga. I roto i tana mahi "La Géométrie," i rongonui a Descartes i te whakamahinga o te pūnaha taunga Cartesian, e āhei ai te whakaahua i ia pūwāhi i roto i te papa mā te takirua tau (x, y).
Ngā Ariā Taketake i roto i te Āhuahanga Tātari
Ngā taunga Cartesian
He mea nui te pūnaha taunga Cartesian ki te āhuahanga tātari. E rua ngā tuaka e hono ana i ngā koki tika i te pūtake (0,0). Ko te tuaka whakapae e mōhiotia ana ko te tuaka-x, ā, ko te tuaka poutū ko te tuaka-y. Mā tēnei pūnaha, ka taea te whakaatu i ia pūwāhi i roto i te papa hei takirua (x, y), ko x te tawhiti whakapae mai i te pūtake, ā, ko y te tawhiti poutū.
Whārite Raina Tika
Ko tētahi o ngā whakamahinga taketake o te āhuahanga tātari ko te whārite o te rārangi tika. Ko te āhua tino noa ko y = mx + b, ko m te pikinga (whakahekenga) ā, ko b te taunga-y. Mā tēnei whārite ka taea e tātou te whakaahua me te tātari i ngā āhuatanga o te rārangi mā te whakamahi i te pāngarau anake.
Hei tauira, hei kimi i te pūwāhi e whakawhiti ana i ngā rārangi e rua, ka whakataurite noa tātou i ngā whārite e rua o ngā rārangi, kātahi ka whakaoti i te pūnaha whārite rārangi e hua ake ana.
Ngā Porowhita me ngā Piko Tapawhā
Ka taea te tautuhi i tētahi porowhita i roto i tētahi pūnaha taunga hei huinga o ngā pūwāhi katoa he tawhiti pumau mai i te pūwāhi pokapū. Ko te whārite paerewa o tētahi porowhita me te pokapū kei te pūwāhi (h, k) me te pūtoro r ko:
\[
(x – h)^2 + (y – k)^2 = r^2
\]
He mea nui anō hoki ngā whārite tapawhā i roto i te āhuahanga tātari. Ko te āhua whānui o te whārite tapawhā ko ax² + by² + cx + dy + e = 0. I runga i ngā uara o a me b, ka taea e tēnei whārite te tohu i tētahi porowhita, i tētahi hyperbola, i tētahi parabola rānei.
Ngā Huringa me te Āhuahanga Tātari
Ka taea te ako i ngā panonitanga āhuahanga pērā i te nekehanga, te hurihanga, me te tauine i roto i te anga o te āhuahanga tātari mā te whakamahi i ngā matihiko me ngā whārite taurangi. Hei tauira, ko te panonitanga hurihanga o te pūwāhi (x, y) mā te koki θ e pā ana ki te pūtake ka taea te whakaatu i roto i te āhua matihiko:
\[
\begin{pmatrix}
x' \\
y '
\end{pmatrix}
=
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta me \cos \theta
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
Ngā Whakamahinga o te Āhuahanga Tātari
He whānuitia ngā tono o te āhuahanga tātari i roto i ngā mara maha, mai i te ahupūngao ki te ōhanga.
Ahupūngao me te Hangarau
I roto i te ahupūngao, ka whakamahia te āhuahanga tātari hei whakaahua i te nekehanga o ngā mea. Ka taea te whakatauira i ngā ariā pēnei i te ara, te tere, me te whakaterenga mā te whakamahi i ngā whārite taunga. I roto i te hangarau, ka whakamahia te āhuahanga tātari i roto i te hoahoa hanganga me te tātari ahotea. Hei tauira, ka taea te whakatau i te wā o te korenga o te whakawhiti-wāhanga mā te whakamahi i ngā mātāpono o te āhuahanga tātari.
Pūtaiao Rorohiko
I roto i ngā whakairoiro rorohiko me te hōtaka, ka whakamahia te āhuahanga tātari hei whakaahua i ngā mea, hei whakahaere i ngā pakiwaituhi, hei tukatuka i ngā whakaahua. Ka whakamahia e ngā rauropi whakaputa-toru-ahu ngā mātāpono taunga me ngā panonitanga āhuahanga hei whakahaere i ngā mea i te wāhi mariko.
Ōhanga me ngā Tatauranga
I roto i te ōhanga, ka whakamahia te āhuahanga tātari hei whakaahua i ngā whanaungatanga i waenga i ngā taurangi ōhanga. Hei tauira, ka taea te whakaatu i te tuku me te tono mā te whakamahi i ngā pihi i roto i tētahi kauwhata rua-ahu. I roto i ngā tatauranga, he whakamahinga ngāwari te kauwhata marara o te āhuahanga tātari, ka tuhia ngā raraunga pūwāhi hei aromatawai i te whanaungatanga i waenga i ngā taurangi e rua.
Ngā Painga o te Āhuahanga Tātari
Ko tētahi o ngā painga matua o te āhuahanga tātari ko tōna kaha ki te hono i ngā peka matua e rua o te pāngarau: te ārangi me te āhuahanga. Mā tēnei ka taea te whakamahi i ngā tikanga ārangi whai hua ake hei whakaoti rapanga āhuahanga, me te rerekētanga. Ko te kaha ki te whakamahi i te tātaitai i roto i te tātari āhuahanga hei kimi i ngā horahanga, ngā rōrahi, me ngā pokapū papatipu he putanga nui i puta mai i te āhuahanga tātari.
Ngā Wero me ngā Whanaketanga Anō
Pērā i ngā mara pūtaiao kē atu, kei te anga whakamua te āhuahanga tātari, ā, kei te whanake tonu. Ko te ako i te āhuahanga i roto i ngā āhuatanga e whā, neke atu rānei, he wāhanga tūhuratanga kaha, inā koa i roto i te ahupūngao ariā me te pūtaiao rorohiko. Ko te āhuahanga rorohiko, e whakamahi ana i ngā mātāpono o te āhuahanga tātari me ngā rauropi uaua hei whakaoti rapanga āhuahanga e uru ana ki te nui o ngā raraunga, he tauira tēnei whanaketanga matatau me ngā tono mahi nui.
Whakamutunga
Ko te āhuahanga tātari tētahi o ngā whakatutukitanga nui i roto i te hītori o te pāngarau, e whakakotahi ana, e whakarei ana hoki i ngā peka matua e rua o te pāngarau: te arapū me te āhuahanga. Mā ngā ariā taketake pēnei i te pūnaha taunga Cartesian, ngā whārite o ngā rārangi tika me ngā porowhita, me ngā panonitanga matihiko ka taea e tātou te mārama me te whakaoti rapanga i roto i te whānuitanga o ngā mara. Mai i te ahupūngao ki te ōhanga me te pūtaiao rorohiko, kei te whai tikanga tonu te āhuahanga tātari, ā, kei te whanake tonu hei whakatutuki i ngā wero o te pāngarau hou.
E akiakihia ana ngā kaipānui e hiahia ana ki ētahi atu āhuatanga hangarau kia ruku hohonu atu ki ngā pukapuka ako me ngā akoranga ipurangi e kapi ana i ēnei kaupapa me ngā taipitopito atu, nā te mea ko te mārama ki te āhuahanga tātari he tūāpapa pakari mō te tūhuratanga atu i roto i ngā manga maha o te pūtaiao me te hangarau.
