Ngā Tauira Pātai e Matapaki ana i te Hurihanga i te Papa Cartesian
He kaupapa nui ngā panonitanga i roto i te papa Cartesian i roto i te pāngarau, inā koa te āhuahanga. Kei roto i ēnei panonitanga ngā mahi maha pērā i te nekehanga, te whakaata, te hurihanga, me te whakawhānui, e whai ana ki te neke, ki te whakarerekē rānei i te āhua o tētahi mea i roto i te papa rua-ahu. Ka arotakehia e tēnei tuhinga ētahi tauira raruraru, ā, ka matapakihia hoki e pā ana ki ngā panonitanga i roto i te papa Cartesian.
Ngā Momo Whakawhiti
I mua i te urunga atu ki te tauira raruraru, me arotake tuatahi tātou i ngā momo panonitanga e whai ake nei:
1. Whakamāoritanga (Nekehanga)
Ko te nekehanga o tētahi pūwāhi, o tētahi mea rānei i roto i tētahi paparangi mā tētahi tawhiti i tētahi ahunga. Ka taea te tautuhi i te nekehanga pēnei:
\[
(x, y) \rightarrow (x+a, y+b)
\]
ko \(a\) me \(b\) ngā tawhiti nekehanga whakapae me te poutū.
2. Whakaaroaro
Ko te whakaata ko te whakaata o tētahi pūwāhi, o tētahi mea rānei puta noa i tētahi tuaka, ahakoa ko te tuaka-x, ko te tuaka-y, ko tētahi atu rārangi rānei. Hei tauira, ko te whakaata puta noa i te tuaka-x:
\[
(x, y) \rightarrow (x, -y)
\]
3. Hurihanga (Tītaha)
Ko te hurihanga ko te hurihanga o tētahi pūwāhi, o tētahi mea rānei huri noa i tētahi pūwāhi pokapū mā tētahi koki. Ka taea te whakaatu i tētahi hurihanga whakamuri mā tētahi koki \(\theta\) penei:
\[
(x, y) \pere matau (x \cos \theta – y \sin \theta, x \sin \theta + y \cos \theta)
\]
4. Whakawhanui (Whakanui)
Ko te whānui ko te panonitanga o te rahi o tētahi mea e tētahi tauwehenga tauine. Mena ko te tauwehenga tauine ko \(k\), ka taea te whakaatu i te whānui penei:
\[
(x, y) \rightarrow (kx, ky)
\]
Ngā Pātai Tauira me te Kōrero
Pātai 1: Whakamāoritanga
Pātai:
Mahia he nekehanga i te pūwāhi \(A(2, 3)\) me te nekehanga o te 5 waeine ki te taha matau, me te 4 waeine ki runga.
Kōrero:
Ko te whakamāori i ngā waeine \(5\) ki te taha matau ko te whakanui ake i te taunga-x mā te \(5\). Ko te kīanga "4 waeine ki runga" ko te whakanui ake i te taunga-y mā te \(4\). Ko te hua o te whakamāoritanga ko:
\[
(x, y) \rightarrow (x+5, y+4)
\]
Nō reira, ko te pūwāhi \(A(2, 3)\) i muri i te whakamāoritanga ka noho hei:
\[
(x+5, y+4) \pere (2+5, 3+4) \pere (7, 7)
\]
Nō reira, ko te pūwāhi \(A(2, 3)\) i muri i te whakamāoritanga ko \(A'(7, 7)\).
Pātai 2: Whakaaroaro
Pātai:
Whakaatahia te pūwāhi \(B(-4, 7)\) mō te tuaka-y.
Kōrero:
Mā te whakaata i te tuaka-y ka huri te taunga-x ki te kino o te taunga-x taketake, ka noho tonu te taunga-y.
\[
(x, y) \rightarrow (-x, y)
\]
Nō reira, ko te pūwāhi \(B(-4, 7)\) i muri i te whakaata ka noho hei:
\[
(x, y) \pere (-(-4), 7) \pere (4, 7)
\]
Nō reira, ko te pūwāhi \(B(-4, 7)\) i muri i te whakaata i te tuaka-y ko \(B'(4, 7)\).
Pātai 3: Hurihanga
Pātai:
Hurihia te pūwāhi \(C(1, 2)\) mā te \(90^\circ\) ki te taha maui me te pokapū kei te pūtake \((0, 0)\).
Kōrero:
Ka taea te whakaatu i te hurihanga o te \(90^\circ\) ki te taha whakamuri penei:
\[
(x, y) \rightarrow (-y, x)
\]
Nō reira, ko te pūwāhi \(C(1, 2)\) i muri i te hurihanga ka rite ki:
\[
(x, y) \rightarrow (-2, 1)
\]
Nō reira, ko te pūwāhi \(C(1, 2)\) i muri i te hurihanga o \(90^\circ\) ki te taha whakamuri ko \(C'(-2, 1)\).
Pātai 4: Whakawhanui
Pātai:
Whakawhanuihia te pūwāhi \(D(3, 4)\) mā te whakamahi i te tauine \(k = 2\).
Kōrero:
Ko te whakawhanuitanga me te tauine tauine o \(2\) ko te tikanga o te whakarea i ngā taunga e rua ki \(2\).
\[
(x, y) \rightarrow (2x, 2y)
\]
Nō reira, ko te pūwāhi \(D(3, 4)\) i muri i te whakawhanuitanga ka noho hei:
\[
(x, y) \rightarrow (2 \times 3, 2 \times 4) \rightarrow (6, 8)
\]
Nō reira, ko te pūwāhi \(D(3, 4)\) i muri i te whakawhanuitanga me te tauine tauine \(2\) ko \(D'(6, 8)\).
Pātai 5: Te Huinga o ngā Huringa
Pātai:
Whakaarohia te tuaka-x, kātahi ka whakawhanuitia mā te whakamahi i te tauine \(k = 0.5\) i te pūwāhi \(E(8, -6)\).
Kōrero:
Ko te taahiraa tuatahi ko te whakaaroaro mō te tuaka-x:
\[
(x, y) \rightarrow (x, -y)
\]
\[
(8, -6) \rightarrow (8, 6)
\]
Ko te taahiraa tuarua ko te mahi i tētahi whakawhanuitanga me te tauine tauine o \(0.5\):
\[
(x, y) \rightarrow (0.5x, 0.5y)
\]
\[
(8, 6) \rightarrow (0.5 \times 8, 0.5 \times 6) \rightarrow (4, 3)
\]
Nō reira, ko te pūwāhi \(E(8, -6)\) i muri i te whakaata mō te tuaka-x me te whakawhānui mā te tauine tauine \(0.5\) ko \(E'(4, 3)\).
Pātai 6: Te Hurihanga me te Hurihanga me te Whakawhiti
Pātai:
Hurihia te pūwāhi \(F(-3, 4)\) mā te \(180^\circ\) ki te taha maui, kātahi ka whakamāoritia te hua me te ira \((2, -1)\).
Kōrero:
Ko te taahiraa tuatahi ko te hurihuri mā te \(180^\circ\) ki te taha maui:
\[
(x, y) \rightarrow (-x, -y)
\]
\[
(-3, 4) \rightarrow (3, -4)
\]
Ko te taahiraa tuarua ko te mahi i tētahi whakamāoritanga me te whārite \((2, -1)\):
\[
(x, y) \rightarrow (x+2, y-1)
\]
\[
(3, -4) \rightarrow (3+2, -4-1) \rightarrow (5, -5)
\]
Nō reira, ko te pūwāhi \(F(-3, 4)\) i muri i te hurihanga \(180^\circ\) me te whakawhiti mā te whārite \((2, -1)\) ko \(F'(5, -5)\).
Whakamutunga
He ariā nui ngā panonitanga i roto i te papa Cartesian i roto i te āhuahanga, e kapi ana i ngā mahi maha pēnei i te nekehanga, te whakaata, te hurihanga, me te whakawhānui. Mā te mārama ki te mahi a ia momo panonitanga, ka taea e tātou te whakarerekē ngāwari i te tūranga, i te āhua rānei o tētahi mea i roto i te papa. Mā roto i ngā tauira i runga ake nei, ka taea e tātou te kite i ngā tono mahi o ngā panonitanga maha me te pehea e taea ai te whakakotahi i a rātou hei whakatutuki i ngā panonitanga uaua ake. Ko te tumanako, kua whai hua tēnei tuhinga ki te mārama ki ngā panonitanga i roto i te papa Cartesian.
