Ngā Huringa i te Papa Cartesian
He ariā taketake te papa Cartesian i roto i te pāngarau me te āhuahanga, e mōhiotia whānuitia ana e ngā ākonga me ngā tohunga pāngarau o te ao katoa. Mā te whakamahi i te pūnaha taunga i whakaurua mai e René Descartes i te rautau 17, ka taea e te papa Cartesian te tuhi kauwhata me te tātari i ngā mahi me ngā āhua āhuahanga i roto i te wāhi rua-ahu. Ko tētahi ariā nui i roto i te tātari āhuahanga o te papa Cartesian ko te panonitanga. I roto i tēnei tuhinga, ka ruku hohonu atu mātou ki ngā momo panonitanga i roto i te papa Cartesian, tae atu ki ngā nekehanga, ngā hurihanga, ngā whakaata, me ngā whakawhānuitanga.
1. Whakamāoritanga
Ko te whakawhiti he momo whakawhiti e neke ana i ia pūwāhi o tētahi mea mā te tawhiti kotahi, me te ahunga kotahi. I roto i te papa Cartesian, ka taea te whakaatu i tētahi whakawhiti mā te whakamahi i tētahi weketore. Hei tauira, ki te whakawhitia tētahi pūwāhi P(x, y) e te weketore (a, b), ka noho te pūwāhi hou P' ki ngā taunga (x + a, y + b). He mea nui te whakawhiti i roto i te whānuitanga o ngā tono, mai i te whakairoiro rorohiko ki te tātari i te nekehanga i roto i te ahupūngao.
Hei tauira, ki te whakawhitia te pūwāhi P(2, 3) e te whārite (4, -1), ka noho te pūwāhi P' ki ngā taunga (6, 2). Mā tēnei panonitanga ka tiakina te āhua me te rahi o te mea, engari ka whakarerekētia tōna tūranga.
2. Hurihanga
Ka hurihia e te hurihanga ia pūwāhi o tētahi mea huri noa i tētahi pūwāhi pokapū kua whakaritea mā tētahi koki motuhake. I roto i te papa Cartesian, ka mahia te hurihanga huri noa i te pūtake (0, 0). Ka taea te whakaatu i te hurihanga hei koki e inehia ana i roto i ngā radian, i ngā nekehanga rānei.
Ko te tātai whānui mō te hurihanga o tētahi pūwāhi P(x, y) mā te koki θ e pā ana ki te pūtake (0, 0) ko:
\[P'(x', y') = (x \cos \theta – y \sin \theta, x \sin \theta + y \cos \theta)\]
Me kī tātou e hiahia ana ki te huri i te pūwāhi P(1, 0) mā te 90 nekehanga ki te taha matau. Mā te whakamahi i te tātai hurihanga:
\\[P'(x', y') = (1 \cos 90° – 0 \sin 90°, 1 \sin 90° + 0 \cos 90°)\]
Ko te hua ko P'(0, 1).
Ko te hurihanga he panonitanga e pupuri ana i te āhua me te rahi o tētahi mea engari e whakarerekē ana i tōna aronga.
3. Whakaaroaro
Ko te whakaata he panonitanga e whakaata ana i ia pūwāhi o tētahi mea e pā ana ki tētahi rārangi tohutoro motuhake. Ko te rārangi tohutoro ka taea te rārangi-x, te rārangi-y, ngā rārangi y = x me y = -x rānei, ētahi atu rārangi rānei.
Mehemea ko te rārangi whakaata ko te tuaka-x, mā te whakaata o te pūwāhi P(x, y) ki te tuaka-x ka puta he pūwāhi P' kei ngā taunga (x, -y).
Ki te whakaatahia te pūwāhi Q(3, 4) puta noa i te tuaka-y, ko ngā taunga o te whakaata Q' ka puta ko (-3, 4). Ka hurihia e te whakaata te aronga o tētahi mea engari ka mau tonu te āhua me te rahi o te mea.
4. Whakawhanui
Ko te whakawhanui he panonitanga e whakanui ana, e whakaiti ana rānei i te rahi o tētahi mea mā tētahi ōwehenga, e pā ana ki tētahi pūwāhi pokapū, ko te pūtake (0, 0) te nuinga. Ka tautuhia te whakawhanui e tētahi tauine tauine k.
Mena he nui ake te tauine i te 1, ka nui ake te mea, engari ki te iti iho te tauine i te 1, ka whakaiti te mea. Ko te tātai whānui ko:
\[ P'(x', y') = (kx, ky) \]
Hei tauira, ki te whakahaerehia he whakawhanuitanga ki te pūwāhi R(2, 3) me te tauine tauine o te 2:
R'(x', y') = (2 \cdot 2, 2 \cdot 3) = (4, 6) \]
Mā tēnei whakawhanuitanga ka whakanuia te tawhiti o tētahi pūwāhi mai i te pūtake mā tētahi tauwehe kua tohua, ā, ka whakarerekē i te rahi whānui o te mea, engari ka mau tonu te āhua taketake o te mea.
Taupānga Whakawhiti
He whānuitia ngā whakamahinga o ngā panonitanga i roto i te papa Cartesian i roto i ngā momo mara o te pūtaiao me te hangarau. I roto i ngā whakairoiro rorohiko, ka whakamahia ngā panonitanga āhuahanga hei whakahaere i ngā whakaahua me ngā mea toru-ahu i runga i te mata rorohiko. Hei tauira, i roto i te pakiwaituhi, ka whakamahia ngā panonitanga pēnei i te nekehanga me te hurihanga hei whakatauira i te nekehanga.
I roto i te ao ahupūngao, ka whakamahia ngā panonitanga hei tātari i te nekehanga o ngā mea. Mā ngā panonitanga taunga ka māmā ake te tatau i ngā ara, i ngā huringa rānei o te tūranga o ngā mea i te wāhi. I roto i te hangarau karetao, ka āwhina ngā panonitanga ki te whakarite i ngā nekehanga me te whakatere o ngā karetao.
I roto i te hangarau ā-iwi me te hoahoanga, ka āwhina ngā panonitanga ā-ira i te hoahoa me te tātari i ngā hanganga whare, tae atu ki te tukanga whakaputa tauira 3D.
He maha ngā wā ka whakamahia e ngā tohunga pāngarau me ngā miihini ngā panonitanga hei mārama ake ki ngā āhuatanga pumau o ngā mea āhuahanga. Ka āwhina tēnei ki te whakamatau i ētahi āhuatanga āhuahanga, ā, ka taea e ngā kaiwhakamahi te whakaoti rapanga uaua ake i roto i te pāngarau tono.
Te Katinga
He taputapu kaha ngā panonitanga i roto i te papa Cartesian mō te tātari me te whakahaere i te āhua me te tūranga o ngā mea i roto i te wāhi rua-ahu. Mā te mārama ki ngā ariā taketake pēnei i te nekehanga, te hurihanga, te whakaata, me te whakawhānui, ka taea e tātou te maioha ki te ataahua pāngarau o te āhuahanga me ōna tono i roto i ngā mara maha o te pūtaiao me te hangarau. Ehara i te mea ko ēnei panonitanga anake te huarahi hei tiro i tō tātou ao kia whai hanganga ake, engari ka taea hoki te tono i taua mōhiotanga i roto i te whānuitanga o ngā auahatanga hangarau me te pūtaiao.
