Ngā Hurihanga Pāngarau: Te Whakarato i tētahi Māramatanga Hurihuri mō te Āhuahanga
Pendahuluan
I roto i te pāngarau, ko te hurihanga tētahi o ngā panonitanga tino taketake me te mea nui, inā koa i roto i te āhuahanga. Ehara i te mea he tono noa iho te hurihanga i roto i te pāngarau parakore engari he pānga hohonu anō hoki ki te pūtaiao me te miihini. Ko te whāinga o tēnei tuhinga he tūhura i te ariā pāngarau o te hurihanga, me pēhea tana mahi, ōna mātāpono matua, me ōna tono o te ao tūturu.
Te Mārama ki te Hurihanga
I roto i te pāngarau, ko te hurihanga e pā ana ki te nekehanga o tētahi mea huri noa i tētahi pūwāhi, tuaka rānei mā tētahi koki motuhake. E mōhiotia ana tēnei pūwāhi, tuaka rānei, ko te pokapū hurihanga. Ina huri tētahi mea, ka neke ia pūwāhi i runga i te mea i te ara porowhita me tētahi pokapū pumau.
Te Tuhituhi me te Kupu Whakataki
I mua i te haere tonu, me mārama ētahi tohu me ngā kupu:
– (x, y) : Ngā taunga Kāreti o tētahi pūwāhi i roto i tētahi papa rua-ahu.
– O : Te pokapū hurihuri.
– θ (theta): Te rahi o te koki hurihuri, e inehia ana i roto i ngā nekehanga, i ngā rāti rānei.
– R(θ, O) : Te mahi hurihanga e tohu ana i te hurihanga mā te koki θ huri noa i te pokapū O.
Tātai Hurihanga i ngā Ahu e Rua
Ka taea te whakaatu i ngā hurihanga mā te whakamahi i ngā matihiko whakawhiti, inā koa i roto i te pūnaha taunga Cartesian rua-ahu. Me kī tātou e hiahia ana ki te huri i te pūwāhi (x, y) mā te koki θ e pā ana ki te pūtake (0, 0). Ka taea te tatau i ngā taunga hou (x', y') i muri i te hurihanga mā te whakamahi i te tātai:
""
x' = x cos(θ) – y sin(θ)
y' = x sin(θ) + y cos(θ)
""
Ka taea tēnei te whakaatu i roto i te puka matihiko penei:
""
| x' | | cos(θ) -hara(θ) | | x |
| koe' | = | hara(θ) cos(θ) | | y |
""
Tauira tauira
Me titiro tātou ki tētahi tauira tuturu hei whakamārama i tō tātou māramatanga. Me kī tātou e hiahia ana ki te huri i te pūwāhi A(1, 0) mā te 90 nekehanga ki te taha maui o te karaka huri noa i te pūtake (0, 0).
""
x' = 1 cos(90°) – 0 hara(90°) = 0
y' = 1 hara(90°) + 0 cos(90°) = 1
""
Nō reira, ko ngā taunga hou o A i muri i te hurihanga ko A'(0, 1).
Te Hurihanga i ngā Ahu-Toru
He uaua ake ngā hurihanga i ngā taha e toru nā te mea he hurihanga huri noa i ngā tuaka x, y, z rānei. Ko ngā matihiko hurihanga i te 3D mō ēnei tuaka e toru koia ēnei:
– Te hurihanga whakarara ki te tuaka X:
""
| 1 0 0 |
| 0 cos(θ) -sin(θ) |
| 0 sin(θ) cos(θ) |
""
– Te hurihanga whakarara ki te tuaka Y:
""
| cos(θ) 0 sin(θ) |
| 0 1 0 |
| -sin(θ) 0 cos(θ) |
""
– Te hurihanga whakarara ki te tuaka Z:
""
| cos(θ) - sin(θ) 0 |
| sin(θ) cos(θ) 0 |
| 0 0 1 |
""
Taupānga Hurihanga Pāngarau
He whānuitia ngā whakamahinga o te hurihanga i roto i ngā momo marautanga. Ko ētahi tauira ko:
Whakairoiro Rorohiko me te Pakiwaituhi
I roto i ngā whakairoiro rorohiko, he maha ngā wā ka whakamahia te hurihanga hei whakahaere me te whakaoho i ngā mea i roto i te wāhi toru-ahu. He mea nui tēnei tikanga mō te waihanga i ngā pānga tirohanga tūturu i roto i ngā kēmu ataata me ngā kiriata pakiwaituhi.
Karetao
I roto i te hangarau karetao, he mea nui te hurihanga hei whakahaere i te nekehanga o te ringa karetao. Mā te whakamahi i ngā panonitanga hurihanga, ka taea e tātou te whakatau i te tūranga whakamutunga me te aronga o te pito-whakahaere o te karetao i muri i te raupapa o ngā nekehanga.
Āhuahanga Ngota
I roto i te matū me te koiora, ka whakamahia ngā hurihanga hei whakatauira i ngā hanganga ngota i roto i ngā taha e toru. Ka taea te tātari me te whakahaere i ngā hanganga ngota mā te whakamahi i ngā hurihanga hurihanga hei mārama ki ngā taunekeneke matū me ngā tauhohenga.
Ahupūngao
I roto i te ahupūngao, he wāhanga nui te hurihanga o ngā āhuatanga maha, tae atu ki ngā whanaketanga tinana-mārō me te miihini irahiko. Hei tauira, ko te wā o te korehau me te nekehanga koki he ariā e pā ana ki te hurihanga.
Whakatere me te Mahere
Ka whakamahia hoki e ngā pūnaha whakatere me te mahere te ariā o te hurihanga. I roto i te GPS, ka whakamahia ngā panonitanga hurihanga hei huri i ngā taunga ā-ao ki ngā taunga ā-rohe hei whakatau tika i te tūranga.
Whakaaturanga me te Whakatauira
He maha ngā wā ka whakamahia he pūmanawa rorohiko hei whakaatu i te hurihanga. He maha ngā pūmanawa, pērā i a MATLAB, GeoGebra, me Python me ngā whare pukapuka pērā i a Matplotlib, Pygame rānei, ka taea te whakamahi hei whakatauira i te hurihanga i roto i ngā āhuahanga e rua, e toru rānei.
Tauira Waehere Python mō te Hurihanga 2D
Anei tētahi tauira māmā i roto i te Python mō te huri i tētahi pūwāhi i roto i ngā āhuatanga e rua:
"`Pitoni
kawemai numpy hei np
kawemai matplotlib.pyplot hei plt
Mahi hei huri i tētahi pūwāhi
def huri_ira(x, y, theta):
teta = np.deg2rad(teta)
x_hou = x np.cos(teta) – y np.sin(teta)
y_hou = x np.sin(teta) + y np.cos(teta)
whakahokia mai x_hou, y_hou
Te tīmatanga (1, 0)
x, y = 1, 0
Hurihanga 90 tohu
teta = 90
pirau_x, pirau_y = pūwāhi_hurihuri(x, y, theta)
Whakaaturanga
plt.ahua()
plt.plot([0, x], [0, y], 'r-', tapanga='Taketake')
plt.plot([0, x_rot], [0, y_rot], 'b-', tapanga='Kua Hurihia')
plt.poutohu()
plt.xlim(-1.5, 1.5)
plt.ylim(-1.5, 1.5)
plt.xlabel('X')
plt.ylabel('Y')
plt.title('Te Hurihanga 90 Ngā Tohu')
whatunga.plt()
plt.whakaatu()
""
E whakaahua ana tēnei waehere i te hurihanga o te pūwāhi (1,0) mā te 90 nekehanga ki te taha maui.
Whakamutunga
He ariā taketake engari he ariā kaha te hurihanga i roto i te pāngarau, inā koa ko te āhuahanga. He mahi nui tana i roto i te whānuitanga o ngā tono o te ao tūturu, mai i ngā whakairoiro rorohiko ki te karetao me te ahupūngao. Mā te mārama ki te mahi a te hurihanga, me te whakaatu pāngarau hoki mā roto i ngā matihiko, ka taea e te tangata te mahi ngāwari i ngā huringa āhuahanga uaua.
Ko te tikanga, mā te hurihanga pāngarau ka huaki te kuaha ki te mārama me te whakahaere i te ao toru-ahu e noho nei tātou, ā, he kaupapa tino nui tēnei mā ngā ākonga, ngā kairangahau, me ngā tohunga ngaio i ngā momo mara.
