Mienzaniso yeMatambudziko ekukurukura nezvekutenderera kwemasvomhu
Pendauluan
Kutenderera kushanduka kwejometri kunowanzoonekwa mumasvomhu, kunyanya geometry. Kutenderera kunosanganisira kutenderedza chinhu chakatenderedza nzvimbo chaiyo (pakati pekutenderera) kuburikidza nekona chaiyo munzira yewachi kana kuti counterclockwise. Pfungwa yekutenderera yakakosha muminda yakaita semakombiyuta, fizikisi, uye mainjiniya. Chinyorwa chino chichaongorora mienzaniso yakasiyana-siyana nehurukuro dzekutenderera mumasvomhu.
Kunzwisisa Kutenderera
Kutenderera kushanduka kunofambisa poindi yega yega yechinhu nekuchitenderedza panzvimbo yakatarwa, inonzi centre of rotation, ne certain angle mune imwe nzira. Runyoro rwekutenderera ne angle θ ne centre of rotation (a, b) runogona kunyorwa se R_(a, b)(θ).
Kune poindi P(x, y) inotenderedzwa ne θ degrees ine pakati pekutenderera pakutanga (0, 0), ma coordinates matsva eP' (x', y') mushure mekutenderera achawanikwa uchishandisa fomura:
– x' = x cos θ – y chivi θ
– y' = x sin θ + y cos θ
Ngatienderere mberi nemimwe mienzaniso yematambudziko ekutenderera azere nehurukuro dzavo.
Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso Mubvunzo 1
Mubvunzo: Sarudza macoordinates matsva epoindi A(3, 4) mushure mekuitenderedza nemadhigirii makumi mapfumbamwe nehafu uchitenderedza nepakati pekutenderera pakutanga (0, 0).
Kukurukurirana: Kushandisa fomura yekutenderera ine angle ye90 degrees counterclockwise:
– x' = x cos(90°) – y chivi(90°) = 3(0) – 4(1) = -4
– y' = x chivi(90°) + y cos(90°) = 3(1) + 4(0) = 3
Saka, makoronesheni matsva eA' mushure mekutenderera ndeaya (-4, 3).
Muenzaniso Mubvunzo 2
Mubvunzo: Poindi B(2, -1) inotenderedzwa nemadhigirii zana nemakumi masere nenzira yewachi, pakati pekutenderera pachiripo pakutanga (0, 0). Sarudza macoordinates matsva epoindi B mushure mekutenderera.
Kukurukurirana: Kutenderera kwemadhigirii zana nemakumi masere kungava kwewachi kana kuti kwewachi kunopa mhedzisiro imwechete, kureva kuti macoordinates epoindi anochinja kuita (-x, -y).
– x' = -x = -2
– y' = -y = 1
Saka, makoroneti matsva eB' ndeaya (-2, 1).
Muenzaniso Mubvunzo 3
Mubvunzo: Poindi C(-3, 5) inotenderedzwa nemadhigirii 270 counterclockwise nepakati pekutenderera panotanga (0, 0). Sarudza poindi C mushure mekutenderera.
Kukurukurirana: Kutenderera kwemadhigirii 270 newachi kwakaenzana nekutenderera kwemadhigirii 90 newachi.
– x' = x cos(90°) + y chivi(90°) = -3(0) + 5(1) = 5
– y' = -x chivi(90°) + y cos(90°) = -(-3)(1) + 5(0) = 3
Saka, ma coordinates matsva eC' mushure mekutenderera ndeaya (5, -3).
Muenzaniso Mubvunzo 4
Mubvunzo: Sarudza macoordinates matsva epoindi D(5, 5) mushure mekuitenderedza nemadhigirii makumi mana nemashanu nepakati pekutenderera pakutanga (0, 0).
Kukurukurirana: Kushandisa fomura yekutenderera ine angle ye45 degrees:
– x' = x cos(45°) – y chivi(45°) = 5(cos(45°)) – 5(chivi(45°)) = 5(√2/2) – 5(√2/2) = 0
– y' = x chivi(45°) + y cos(45°) = 5(√2/2) + 5(√2/2) = 5√2/2 + 5√2/2 = 5√2
Saka, ma coordinates matsva eD' mushure mekutenderera ndeaya (0, 5√2).
Kutenderera neCenter of Rotation Kwete paKutanga
Kutenderera hakugaroitwa maererano nekwakabva. Semuenzaniso, ngatitii tinoda kutenderera poindi ine pakati pekutenderera (h, k). Kuti tiite izvi, tinofanira kugadzirisa macoordinates seizvi:
1. Shandura pfungwa yacho kuitira kuti (h, k) ive mavambo.
2. Shandisa fomura yekutenderera.
3. Dudzira zvakare kunzvimbo yekutanga.
Muenzaniso Mubvunzo 5
Mubvunzo: Poindi E(5, 7) inotenderedzwa nemadhigirii makumi mapfumbamwe nehafu zvichipesana newachi nepakati pekutenderera panzvimbo (2, 3). Sarudza macoordinates matsva epoindi E mushure mekutenderera.
Kukurukurirana:
1. Shandura poindi E kuenda kumavambo zvichienderana nepakati pekutenderera (2, 3):
– Poindi itsva E' = (5 – 2, 7 – 3) = (3, 4)
2. Tenderedza madhigirii makumi mapfumbamwe uchitenderedza nzvimbo itsva:
– x' = 3 cos(90°) – 4 chivi(90°) = -4
– y' = 3 chivi(90°) + 4 cos(90°) = 3
Saka, macoordinates mushure mekutenderera ndeaya (-4, 3).
3. Kushandura kudzokera kunzvimbo yekutanga kana tichienzanisa nepakati pekutenderera (2, 3):
– Pokupedzisira E' = (-4 + 2, 3 + 3) = (-2, 6)
Saka, makorodheni matsva epoindi E mushure mekutenderera ndeaya (-2, 6).
Mhedziso
Kuongorora nekunzwisisa kutenderera kwemasvomhu kwakakosha mukushandiswa kwakasiyana-siyana. Kuburikidza nemienzaniso nehurukuro dziri pamusoro apa, vaverengi vanotarisirwa kunzwisisa kuti mafomula ekutenderera anoshanda sei uye kuti angashandiswa sei mumamiriro akasiyana. Chiitwa ichi hachingosimbisi chete masvomhu ekutanga asiwo chinobatsira mune dzimwe nzvimbo dzinosanganisira kushanduka kwejometri.
