# ध्रुवीय निर्देशांक पद्धति

गणित में ध्रुवीय निर्देशांक पद्धति (polar coordinate system) द्विविमीय-निर्देशांक पद्धति है जिसमें किसी बिन्दु के निर्देशांक उस बिन्दु की किसी सन्दर्भ बिन्दु से दूरी एवं सन्दर्भ दिशा से बनने वाले कोण द्वारा दी जाती है।

## ध्रुवीय एवं कार्तीय निरेशांकों में परस्पर परिवर्तन

A diagram illustrating the relationship between polar and Cartesian coordinates.
A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, ${\displaystyle y=\sin(6x)+2}$ is mapped onto ${\displaystyle r=\sin(6\varphi )+2}$. Click on image for details.

The polar coordinates r and ϕ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine:

${\displaystyle x=r\cos \varphi \,}$
${\displaystyle y=r\sin \varphi \,}$

The Cartesian coordinates x and y can be converted to polar coordinates r and ϕ with r ≥ 0 and ϕ in the interval (−π, π] by:[1]

${\displaystyle r={\sqrt {x^{2}+y^{2}}}\quad }$ (as in the Pythagorean theorem or the Euclidean norm), and
${\displaystyle \varphi =\operatorname {atan2} (y,x)\quad }$,

where atan2 is a common variation on the arctangent function defined as

${\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0\end{cases}}}$

The value of ϕ above is the principal value of the complex number function arg applied to x+iy. An angle in the range [0, 2π) may be obtained by adding 2π to the value in case it is negative.

## सन्दर्भ

1. Torrence, Bruce Follett; Eve Torrence (1999). The Student's Introduction to Mathematica. Cambridge University Press. आई॰ऍस॰बी॰ऍन॰ 0-521-59461-8.