Makorodheni ePolar muGeometry
Mu geometry, kuti "tinotumidza" sei nzvimbo yepoindi zvinonyanya kutsanangura kuti tinonzwisisa sei maumbirwo, madaro, maenga, uye hukama pakati pezvinhu. Sisitimu ye coordinate inozivikanwa zvikuru ndiyo Cartesian coordinate system, iyo inoshandisa pair \((x, y)\) kumiririra nzvimbo yepoindi iri mundege. Zvisinei, kune imwe sisitimu inowanzo kuve yakajairika mumamiriro ezvinhu anosanganisira madenderedzwa, kutenderera, mafambiro, uye madaro kubva pakati: polar coordinate. Chinyorwa chino chinokurukura pfungwa ye polar coordinate, maitiro ekuiverenga, hukama hwayo neCartesian coordinate, uye mamwe mashandisirwo mu geometry.
1. Kunzwisisa Polar Coordinates
Polar coordinates isystem ine ma coordinate maviri-dimensional inomiririra poindi yakavakirwa pa:
1. Kureba kwenzvimbo kubva pakati (kwakabva) kunonzi radius uye kunomiririrwa ne \(r\).
2. Kona yekutungamira kwenzvimbo kuenda kumucheto wereferensi, kazhinji kuenda kumuganhu we \(x\) wakanaka, inonzi kona yepolar uye inoratidzwa ne \(\theta\).
Saka, nzvimbo yepoindi mu polar coordinates inonyorwa se \((r, \theta)\).
– \(r\) inotaura kuti "pfungwa yacho iri kure zvakadii" kubva pakati.
– \(\theta\) inoratidza "kwairi" poindi, ichiyerwa sekona kubva pakatarisana kuenda kurudyi (positive \(x\) axis) kuenda kunzvimbo yepoindi, kazhinji zvichienderana newachi.
Semuenzaniso, poindi \((5, 30^\circ)\) zvinoreva poindi iri kure nemayuniti mashanu kubva pakati uye inoumba kona yemadhigirii makumi matatu kubva pa positive \(x\) axis.
2. Zvinhu Zvikuru: Nzvimbo Yepakati, Akisi, uye Angle
Muma coordinates e polar, pakati pema coordinates inonzi pole (yakaenzana nekwakabva mu Cartesian coordinates). Kubva pa pole, mutsetse wekutarisa kutungamira kwe angle unonzi polar axis, kazhinji unoenderana ne positive \(x\) axis.
Kuyerwa kwemakona \(\theta\) kunogona kuyerwa nemadhigirii kana ma radians. Mumasvomhu epamusoro, ma radians anonyanya kushandiswa nekuti anoita kuti kuverenga kuve nyore:
– \(180^\circ = \pi\) maredhiyani
– \(360^\circ = 2\pi\) radians
Saka kona ye30° yakaenzana ne \(\frac{\pi}{6}\), uye 45° yakaenzana ne \(\frac{\pi}{4}\).
3. Kusiyana kweKumiririrwa kwePoin muPolar Coordinates
Kusiyana nemakotesheni eCartesian, poindi imwe chete mumakotesheni epolar inogona kuva nemifananidzo inopfuura imwe chete. Izvi zvinoitika nekuti:
1. Makona anogona kuwedzerwa nekuwedzera kwe \(2\pi\) pasina kuchinja divi.
\[
(r,\theta) \equiv (r,\theta + 2k\pi)
\]
yenhamba huru \(k\).
2. Kukosha kwe \(r\) kunogona kuva negative, zvinoreva kuti poindi iri mudivi rakapesana nekona \(\theta\):
\[
(r,\theta) \equiv (-r, \theta + \pi)
\]
Semuenzaniso, \((3, \frac{\pi}{4})\) inonongedzera panzvimbo imwechete ne \((3, \frac{9\pi}{4})\) nekuti maangles anosiyana nekuchinja kwakazara kamwe chete. Nzvimbo imwe chete inogonawo kuratidzwa se \((-3, \frac{5\pi}{4})\).
Zvakakosha kunzwisisa kusiyana uku kuitira kuti usavhiringidzwa paunenge uchigadzirisa ma equation mu polar coordinates.
4. Kuchinja pakati pePolar neCartesian Coordinates
Chimwe chezvinhu zvakakosha pakudzidza ma polar coordinates kunzwisisa maitiro ekushandura kuita ma Cartesian coordinates uye zvinopesana nazvo. Hukama hwacho hunobva pa trigonometry muma right triangles.
Kubva ku polar \((r,\theta)\) kuenda kuCartesian \((x,y)\):
\[
x = r\cos\theta
\]
\[
y = r\sin\theta
\]
Kubva kuCartesian \((x,y)\) kuenda ku polar \((r,\theta)\):
\[
r = \sqrt{x^2 + y^2}
\]
\[
\theta = \arctan\left(\frac{y}{x}\right)
\]
Zvisinei, pa \(\theta\) tinofanira kutarisisa quadrant. Sezvo basa re \(\arctan\) rakajairika richingoburitsa maangles pane imwe range, tinowanzo shandisa pfungwa yemaquadrants kana basa \(\text{atan2}(y,x)\) mukuverenga kuti tiwane \(\theta\ chaiyo).
Muenzaniso: Kana poindi \((x,y)=(-1,1)\), ipapo \(\frac{y}{x}=-1\) kuitira kuti \(\arctan(-1)\) ipe \(-45^\circ\), kunyangwe poindi iri mu quadrant II saka kona chaiyo iri \(135^\circ\).
5. Curve Equation muPolar Coordinates
Chimwe chikonzero chinoita kuti polar coordinates ive yakakosha mu geometry ndechekuti maumbirwo mazhinji anova nyore kana akanyorwa mu polar form.
a. Denderedzwa Rakatarisana Nekwakabva
Denderedzwa rine radius \(a\) uye pakati pekwakabva rakareruka kwazvo:
\[
r = a
\]
Izvi zvakapfupika kupfuura chimiro cheCartesian:
\[
x^2 + y^2 = a^2
\]
b. Mutsetse Wakatwasuka Uchipfuura Nekwakabva
Mutsetse unogadzira kona \(\alpha\) kuenda ku \(x\) axis unogona kuratidzwa seizvi:
\[
\theta = \alpha
\]
MuCartesian, mutsetse uyu unova \(y = (\tan\alpha)x\), izvo zvinoenderana nekutsetseka kwawo.
c. Makombama uye Makombama Akakosha
Mamwe ma curves anoshanda anoratidzwa mu polar, semuenzaniso:
– Kutenderera kwaArchimedes: \(r = a\theta\)
– Mwoyo: \(r = a(1+\cos\theta)\)
– Limaçon : \(r = a + b\cos\theta\)
– Ruva (ruvara rweruvara): \(r = a\cos(k\theta)\) kana \(r = a\sin(k\theta)\)
Makomba aya anowanzoonekwa muhurukuro dze geometry, graphics, uye physics.
6. Kureba uye Angle muPolar Coordinates
Nekuti polar coordinates yakavakirwa pa radius ne angle, mamwe maverengero e geometric anova nyore kunzwisisa. Semuenzaniso, daro kubva pane imwe poindi kuenda kune kwakabva rinopihwa zvakananga na \(r\). Kune daro riri pakati pemapoinzi maviri \((r_1,\theta_1)\) uye \((r_2,\theta_2)\), tinogona kushandisa mutemo we cosines:
\[
d^2 = r_1^2 + r_2^2 – 2r_1r_2\cos(\theta_1 – \theta_2)
\]
Fomura iyi inobatsira zvikuru kana pfungwa mbiri dzichitsanangurwa se "daro kubva pakati" uye musiyano wegwara, semuenzaniso mumatambudziko ane chekuita nezvikamu zvedenderedzwa kana magadzirirwo eradial.
7. Kushandiswa kwePolar Coordinates muGeometry neHupenyu Hwechokwadi
Makoroniti ePolar haasi kungori pfungwa isina kujeka chete, asiwo ane mashandisirwo akawanda chaiwo:
1. Kufamba nemepu: nzvimbo inogona kuratidzwa sedaro uye gwara kubva panzvimbo yekunongedzera.
2. Kuona nyeredzi: nzvimbo yezvisikwa zvemuchadenga inowanzo tsanangura kona yemutsetse wekutarisa uye daro rakati.
3. Robotics nemasensa: radar neLIDAR zvinowanzo gadzira data muchimiro chemadaro nemakona, izvo zviri polar zvakasikwa.
4. Dhizaini yemakombiyuta uye mifananidzo: mapatani akatenderera, maanimation ekutenderera, uye mhedzisiro yemafungu e radial zviri nyore kushanda nazvo mu polar coordinates.
5. Magadzirirwo ezvivakwa nemainjiniya: zvivakwa zvakafanana ne radially symmetrical (madomes, magiya, ma turbine) zvinowanzo ongororwa zviri nyore nezviri polar.
Mu geometry yakachena, polar coordinates inobatsira kunzwisisa denderedzwa symmetry, rotational transformations, uye hukama hwemaumbirwo akatarisana nepfungwa.
8. Kesimpulan
Mapolar coordinates isystem ye coordinates inoratidza nzvimbo yepoindi kuburikidza ne radius \(r\) uye angle \(\theta\). Zvichienzaniswa ne Cartesian coordinates, polar coordinates inopa nzira yakajairika yekuona zvinhu nematambudziko ane chekuita nemadenderedzwa, kutenderera, uye kufamba kwe radial. Nekunzwisisa kushandurwa pakati pe polar coordinates ne Cartesian coordinates, uye kuziva kuti ma equation ema curves anova nyore sei mu polar coordinates, tinowana chishandiso chine simba chekuongorora mamiriro akasiyana-siyana e geometrical.
Pakupedzisira, kugona ma polar coordinates hakusi kungodzidza "imwe nzira yekunyora mapoinzi," asiwo kuwedzera kufunga kwe geometric: kubva kune imwe yakavakirwa pamitsara yakatwasuka kuenda kune imwe yakavakirwa padaro negwara. Izvi zvinoita kuti polar coordinates ive yakakosha mu geometry nedzimwe nzvimbo dzakawanda dzinoshandiswa.
