2D Coordinates Systems

The base / referential change using cartesian coordinates $ (x, y) $ to another referential using polar coordinates $ (r, \theta) $ obey the equations: $$ r = \sqrt{x^2 + y^2} \\ \theta = 2\arctan\left(\frac y{x+ \sqrt{x^2+y^2}} \right) $$ with $ \arctan $ the reciprocal of the function $ \tan $ (tangent).

NB: the value of $ \theta $ calculated here is included in the inverval $ ] -\pi, \pi] $ (to have it in the interval $ ] 0, 2\pi] $ add $ 2 \pi $ if the value of the angle is negative)

If $ r = 0 $ then the angle can be defined by any real number

Example: The point of the plane in position $ (1,1) $ in Cartesian coordinates is defined by the polar coordinates $ r = \sqrt {2} $ and $ \theta = \pi/4 $

The base / referential change from polar coordinates $ (r, \theta) $ to another referential using cartesian coordinates $ (x, y) $ follows the equations: $$ x = r \cos (\theta) \\ y = r \sin (\theta) $$

with $ r $ a positive real number and $ \theta $$ an angle defined between $ ] -\pi, \pi] $