Te Tūnga o ngā Porowhita e Rua: He Tātari Āhuahanga
I roto i te pāngarau, inā koa i roto i te āhuahanga, he mea nui te mārama ki te tūranga o ngā porowhita e rua. Ko ngā porowhita tētahi o ngā āhuahanga taketake e kitea pinepinetia ana i roto i ngā mahi ariā me ngā mahi ā-ringa. Mā te tūranga o ngā porowhita e rua ka mārama ki te taunekeneke o ēnei āhua e rua ina whakatakotoria ki roto i te papa. Kei roto i tēnei rangahau he tātaritanga o ngā taunekeneke ka taea te puta, mai i te kore-whakawhitinga ki te whakawhitinga. Ka arotakehia whānuitia e tēnei tuhinga te tūranga o ngā porowhita e rua me ngā āhuatanga e pā ana.
Ngā Whakamāramatanga me ngā Tuhituhinga
Tuatahi, me tautuhi ā-ture tātou i ngā porowhita e rua i te papa Cartesian. Ka taea te whakaatu i te porowhita \(C_1\) me te pokapū \(P_1(x_1, y_1)\) me te radius \(r_1\) mā te whārite:
\[
C_1 : (x – x_1)^2 + (y – y_1)^2 = r_1^2
\]
Waihoki, ko te porowhita \(C_2\) me te pokapū \(P_2(x_2, y_2)\) me te pūtoro \(r_2\) e tohuhia ana e:
\[
C_2 : (x – x_2)^2 + (y – y_2)^2 = r_2^2
\]
Ko te tūnga o ēnei porowhita e rua e whakawhirinaki ana ki te tawhiti i waenganui i ō rāua pokapū (\(d\)) me te roa o ō rāua radius. Ka taea te tatau i te tawhiti \(d\) i waenganui i ngā pokapū o ngā porowhita e rua \(P_1\) me \(P_2\) mā te whakamahi i te tātai:
\[
d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}
\]
Kāwai Tūnga Porowhita Rua
I te nuinga o te wā, e rima ngā tūranga ka pā ki ngā porowhita e rua:
1. Tūponotanga (Te Taurite o ngā Porowhita e Rua)
2. Kāore e whakawhiti (kāore e wehe tetahi i tetahi)
3. Pānga o Waho
4. Pā ā-roto (Tangent ā-roto)
5. Te Whakawhitiwhiti
He āhuatanga āhuahanga motuhake tō ia o ēnei kāwai, ā, ka matapakihia e mātou ēnei i raro nei.
1. Tūponotanga (Te Taurite o ngā Porowhita e Rua)
E kiia ana he ōrite, he ōrite rānei ngā porowhita e rua mēnā he ōrite te pokapū me te radius. Ko te tikanga o tēnei i roto i te pāngarau:
\[
P_1 \equiv P_2 \quad \text{and} \quad r_1 = r_2
\]
I tēnei wā, \(d = 0\). He ōrite ngā porowhita e rua, ā, ko ia pūwāhi i tētahi porowhita he pūwāhi i tētahi atu porowhita.
2. Kāore e whakawhiti (kāore e wehe tetahi i tetahi)
E kiia ana ngā porowhita e rua kāore e whakawhiti ana i raro i ngā āhuatanga e rua:
– Te Āhuatanga Tuatahi: Ina nui ake te tawhiti i waenganui i ngā pokapū o ngā porowhita e rua (d) i te tapeke o ngā roa o ō rāua rāwhiti:
\[
d > r_1 + r_2
\]
– Āhuatanga Tuarua: Ina kei roto tētahi porowhita i tētahi atu porowhita, kāore e pā ana. Ka puta tēnei mēnā:
\[
d < |r_1 - r_2| \] I ngā āhuatanga e rua, kāore he pūwāhi ōrite i waenganui i ngā porowhita \(C_1\) me \(C_2\). 3. Te Pānga o Waho Ka pānga o waho ngā porowhita e rua mēnā ka pā ki tētahi pūwāhi, ā, kei waho tetahi i tetahi. Ka puta tēnei mēnā he ōrite te tawhiti i waenganui i ngā pokapū o ngā porowhita e rua ki te tapeke o ō rāua radius:
\[ d = r_1 + r_2 \] I tēnei take, kotahi tonu te pūwāhi e rite ana ki te pūwāhi o te pātaki o ngā porowhita e rua. 4. Pātaki ā-Roto He pātaki ā-roto ngā porowhita e rua ina pā tetahi porowhita ki tetahi atu porowhita mai i roto i tētahi pūwāhi kotahi. Ko te tikanga mō tēnei: \[ d = |r_1 - r_2| \] I konei anō hoki, kotahi tonu te pūwāhi o te pātaki, engari kāore i rite ki te pātaki ā-waho, kei roto tetahi porowhita i tetahi atu. 5. Te Whakawhitiwhiti Ka whakawhiti ngā porowhita e rua mēnā he rua ngā pūwāhi o te whakawhitiwhitinga o rāua. Mō tēnei take, ko te tikanga me tutuki ko: \[ |r_1 - r_2| < d < r_1 + r_2 \] I tēnei take, e rua ngā pūwāhi o te whakawhitiwhitinga e tūtaki ai ngā porowhita e rua. Ko tēnei take te mea tino uaua, tino whakamere hoki, nā te mea e rua ngā otinga e pā ana ki te whārite tapawhā e puta mai ana i te pūnaha whārite o ngā porowhita \(C_1\) me \(C_2\). Te Tātari Pāngarau i te Tūnga o ngā Porowhita e Rua Mā te tirotiro hohonu i te tūnga o ngā porowhita e rua, ka whakamahia he huarahi tātari hei mārama ki ngā pūwāhi o te pātata, o te whakawhitiwhiti rānei. Mā te whakaoti i te whārite o ngā porowhita e rua ka puta he pūnaha o ngā whārite tapawhā, ka taea te whakaoti mā te whakakapinga.
Hei tauira, hei kimi i te pūwāhi whakawhiti o ngā porowhita e rua \(C_1\) me \(C_2\), ka tangohia e mātou ngā whārite porowhita e rua hei whakakore i te tapawhā o te taurangi, ka puta he whārite rārangi. Mā te otinga o tēnei whārite rārangi ka puta tētahi o ngā taurangi i runga i te āhua o tētahi atu, ā, mā te whakakapinga ki tētahi o ngā whārite porowhita taketake ka puta te uara o te pūwāhi whakawhiti. Ngā Whakamahinga o te Tūnga o ngā Porowhita e Rua I te ao tūturu, he whānui ngā whakamahinga o te mārama ki te tūranga o ngā porowhita e rua, mai i te hoahoa miihini ki te tātari whatunga. Ka kitea he tauira tūturu i roto i te hoahoa taputapu, he mea nui te pātaki o waho i waenga i ngā porowhita e rua. I roto i te tātari whakawhitiwhiti whatunga, ka whakamahia pinepine te ariā o ngā porowhita hei whakatau i te whānuitanga mōrahi o te tuku tohu. Whakamutunga Mā te tūranga o ngā porowhita e rua ka mārama ki te taunekeneke taketake i waenga i ngā āhua āhuahanga e rua. Ahakoa he māmā noa iho tēnei ariā, he pānga hohonu ki ngā momo mara o te pūtaiao me te miihini. He mea nui kia mārama ngā ākonga me ngā tohunga ki tēnei ariā kia taea ai te whakamahi i ngā mātāpono āhuahanga ki te whakaoti rapanga mahi i roto i te oranga o ia rā. Mai i ngā tūponotanga ki ngā whakawhiti, kei ia tūranga o ngā porowhita e rua ngā mōhiohio nui e whai hua ana mō te tātari me te hoahoa. Mā te mārama ki ngā āhuatanga pāngarau me ngā pānga o ia tūranga ka āwhina i te whakapai ake i te whai huatanga me te whaihua i roto i ngā mahi ā-tinana. Nō reira, ko te ako i te tūranga o ngā porowhita e rua he tūāpapa nui e tautoko ana i te māramatanga whānui o te āhuahanga me te pāngarau whānui.
