Likhokahano tsa Polar ho Geometry
Ho geometry, tsela eo re "rehang" boemo ba ntlha ka eona e laola haholo-holo kamoo re utloisisang libopeho, libaka, likhutlo le likamano lipakeng tsa lintho. Sistimi e tsebahalang haholo ea likhokahano ke sistimi ea likhokahano ea Cartesian, e sebelisang para \((x, y)\) ho emela sebaka sa ntlha sefofaneng. Leha ho le joalo, ho na le sistimi e 'ngoe eo hangata e leng ea tlhaho haholo bakeng sa maemo a amanang le li-circles, lipotoloho, litaelo le libaka tse tsoang bohareng: likhokahano tsa polar. Sengoloa sena se tšohla mohopolo oa likhokahano tsa polar, mokhoa oa ho li bala, kamano ea tsona le likhokahano tsa Cartesian, le lits'ebetso tse ling ho geometry.
1. Ho utloisisa Likhokahano tsa Polar
Likhokahano tsa polar ke sistimi ea likhokahano tsa mahlakore a mabeli e emelang ntlha e thehiloeng ho:
1. Sebaka sa ntlha ho tloha bohareng (mohloli) se bitsoa radius mme se tshwantshwa ke \(r\).
2. Sekhutlo sa tataiso ya ntlha ho ya ho axis ya referense, hangata ho ya ho axis e ntle ya \(x\), se bitswa angle ya polar mme se bontshwa ke \(\theta\).
Kahoo, boemo ba ntlha ho li-coordinate tsa polar bo ngoloa e le \((r, \theta)\).
– \(r\) e bolela “hore na ntlha e hole hakae” le bohareng.
– \(\theta\) e bontsha “ntlha e lebile kae”, e lekantsweng e le sekhutlo ho tloha ho rapameng ho ya ho le letona (mothapo o motle \(x\)) ho ya boemong ba ntlha, ka kakaretso ho ya ka lehlakoreng le leng.
Mohlala, ntlha \((5, 30^\circ)\) e bolela ntlha e bohole ba diyuniti tse 5 ho tloha bohareng mme e bopa sekhutlo sa di-degree tse 30 ho tloha ho axis e ntle \(x\).
2. Dikarolo tsa Motheo: Ntlha e Bohareng, Axis, le Sekhutlo
Ho di-coordinates tsa polar, setsi sa di-coordinates se bitswa palo (e lekanang le tshimoloho ho di-coordinates tsa Cartesian). Ho tloha palong, mola wa referense bakeng sa tataiso ya sekhutlo o bitswa polar axis, hangata o tsamaellanang le positive \(x\) axis.
Litekanyo tsa likhutlo \(\theta\) li ka lekanngoa ka likhato kapa li-radian. Lipalong tse tsoetseng pele, li-radian li sebelisoa haholo hobane li nolofatsa lipalo:
– \(180^\circ = \pi\) mahlasedi
– \(360^\circ = 2\pi\) di-radian
Kahoo sekhutlo sa 30° se lekana le \(\frac{\pi}{6}\), mme 45° e lekana le \(\frac{\pi}{4}\).
3. Ho ikhetha ha Boemeli ba Lintlha ho Li-Polar Coordinate
Ho fapana le likhokahano tsa Cartesian, ntlha e le 'ngoe likhokahanong tsa polar e ka ba le litšoantšo tse fetang e le 'ngoe. Sena se etsahala hobane:
1. Likhutlo li ka eketsoa ka makhetlo a mangata a \(2\pi\) ntle le ho fetola tsela.
\[
(r,\theta) \equiv (r,\theta + 2k\pi)
\]
bakeng sa kakaretso \(k\).
2. Boleng ba \(r\) e ka ba negative, ho bolelang hore ntlha e ka lehlakoreng le fapaneng le angle \(\theta\):
\[
(r,\theta) \equiv (-r, \theta + \pi)
\]
Mohlala, \((3, \frac{\pi}{4})\) e supa ntlha e le 'ngoe le \((3, \frac{9\pi}{4})\) hobane likhutlo li fapana ka phetoho e le 'ngoe e felletseng. Ntlha e tšoanang e ka boela ea hlalosoa e le \((-3, \frac{5\pi}{4})\).
Ho bohlokoa ho utloisisa ho ikhetha hona e le hore o se ke oa ferekana ha o sebetsana le li-equation ka har'a li-coordinate tsa polar.
4. Phetoho pakeng tsa Li-coordinate tsa Polar le Cartesian
E 'ngoe ea likarolo tsa bohlokoa ka ho fetisisa tsa ho ithuta li-coordinates tsa polar ke ho utloisisa mokhoa oa ho li fetolela ho li-coordinates tsa Cartesian le ka tsela e fapaneng. Kamano ena e simoloha ho trigonometry ka likhutlotharo tse nepahetseng.
Ho tloha polar \((r,\theta)\) ho ea ho Cartesian \((x,y)\):
\[
x = r\cos\theta
\]
\[
y = r\sin\theta
\]
Ho tloha ho Cartesian \((x,y)\) ho ea ho polar \((r,\theta)\):
\[
r = \sqrt{x^2 + y^2}
\]
\[
\theta = \arctan\left(\frac{y}{x}\right)
\]
Leha ho le jwalo, bakeng sa \(\theta\) re hloka ho ela hloko quadrant. Kaha mosebetsi o tlwaelehileng wa \(\arctan\) o hlahisa dikhutlo feela hodima sebaka se itseng, hangata re sebedisa mohopolo wa di-quadrant kapa mosebetsi \(\text{atan2}(y,x)\) dipalong ho fumana \(\theta\) e nepahetseng.
Mohlala: Haeba ntlha \((x,y)=(-1,1)\), joale \(\frac{y}{x}=-1\) e le hore \(\arctan(-1)\) e fane \(-45^\circ\), leha ntlha e le ka quadrant II kahoo sekhutlo sa nnete ke \(135^\circ\).
5. Khokahano ea Curve ho Li-Polar Coordinates
Lebaka le leng leo ka lona likhokahano tsa polar li leng bohlokoa ho geometry ke hore libopeho tse ngata li ba bonolo haholo ha li ngoloa ka sebopeho sa polar.
a. Selikalikoe se Behiloeng Bohareng Tšimolohong
Selikalikoe se nang le radius \(a\) le setsi qalong se bonolo haholo:
\[
r = a
\]
Sena se khutšoanyane haholo ho feta sebopeho sa Cartesian:
\[
x^2 + y^2 = a^2
\]
b. Mola o Otlolohileng ho ea Tšimolohong
Mola o etsang sekhutlo sa axis ...
\[
\theta = \alpha
\]
Ka Cartesian, mola ona o fetoha \(y = (\tan\alpha)x\), e leng se itshetlehileng ka leralla la wona.
c. Li-spiral le Li-curve tse Ikhethang
Li-curve tse ling tse sebetsang li hlahisoa ka polar, mohlala:
– Sekoli sa Archimedes: \(r = a\theta\)
– Pelo: \(r = a(1+\cos\theta)\)
– Limaçon : \(r = a + b\cos\theta\)
– Rosa (sekotwana sa rosa): \(r = a\cos(k\theta)\) kapa \(r = a\sin(k\theta)\)
Li-curve tsena hangata li hlaha lipuisanong tsa jeometri, litšoantšo le fisiks.
6. Sebaka le Sekhutlo ho Dikoordinate tsa Polar
Kaha likhokahano tsa polar li thehiloe ho radius le angle, lipalo tse ling tsa jeometri li fetoha tse utloahalang haholoanyane. Mohlala, sebaka ho tloha ntlheng ho ea qalong se fanoa ka ho toba ke \(r\). Bakeng sa sebaka se pakeng tsa lintlha tse peli \((r_1,\theta_1)\) le \((r_2,\theta_2)\), re ka sebelisa molao oa li-cosine:
\[
d^2 = r_1^2 + r_2^2 – 2r_1r_2\cos(\theta_1 – \theta_2)
\]
Foromo ena e thusa haholo ha dintlha tse pedi di hlaloswa ka "sebaka ho tloha bohareng" le phapang tseleng, mohlala mathateng a kenyeletsang dikarolo tsa sedikadikwe kapa dibopeho tsa radial.
7. Tšebeliso ea Li-coordinate tsa Polar ho Geometry le Bophelo ba Sebele
Likhokahano tsa polar ha se feela mohopolo o sa utloisiseheng, empa hape li na le lits'ebetso tse ngata tsa 'nete:
1. Ho tsamaya le ho etsa 'mapa: sebaka se ka hlalosoa e le sebaka le tataiso ho tloha ntlheng ea referense.
2. Bolepi ba linaleli: sebaka sa lihloliloeng tsa leholimo hangata se hlalosa sekhutlo ho ea mola oa litšupiso le sebaka se itseng.
3. Liroboto le li-sensor: radar le LIDAR hangata li hlahisa data ka mokhoa oa libaka le likhutlo, tseo ka tlhaho li leng polar.
4. Moralo le litšoantšo tsa khomphutha: dipaterone tse chitja, di-animation tsa ho potoloha, le ditlamorao tsa maqhubu a radial di bonolo ho sebetsa ka tsona di-coordinate tsa polar.
5. Meaho le boenjiniere: meaho e lekanang ka mahlaseli (matlo, ligiya, liturbine) hangata e hlahlojoa habonolo ka e polar.
Ho jeometri e hloekileng, likhokahano tsa polar li thusa ho utloisisa ho lekana ha selikalikoe, liphetoho tsa potoloho, le likamano tsa libopeho tse shebaneng le ntlha.
8. Kesimpulan
Li-coordinates tsa polar ke sistimi ea li-coordinates e hlalosang sebaka sa ntlha ka radius \(r\) le angle \(\theta\). Ha li bapisoa le li-coordinates tsa Cartesian, li-coordinates tsa polar li fana ka mokhoa oa tlhaho haholoanyane oa ho sheba lintho le mathata a amanang le li-circles, lipotoloho le motsamao oa radial. Ka ho utloisisa phetoho lipakeng tsa li-coordinates tsa polar le Cartesian, le ho lemoha kamoo li-equation tsa li-curve li bang bonolo kateng li-coordinates tsa polar, re fumana sesebelisoa se matla sa ho sekaseka maemo a fapaneng a jeometri.
Qetellong, ho tseba likhokahano tsa polar ha se feela taba ea ho ithuta "mokhoa o mong oa ho ngola lintlha," empa hape le taba ea ho atolosa monahano oa jeometri: ho tloha ho o mong o thehiloeng meleng e otlolohileng ho ea ho o mong o thehiloeng bohōleng le tataisong. Sena se etsa hore likhokahano tsa polar e be tsa bohlokoa jeometring le masimong a mang a mangata a sebelisitsoeng.
