Required Information

Two conditions ore required to specify the locotion of a point relative to an origin in two$-$dimensional space $($as show in the figure$).$

The horizontal and vertical distances $(x,y)$ in Cartesion coordinates.

The radius and angle $(r,)$ in polar coordinates.

It is relotively strsightforward to compute Cartesian coordinates bosed on polar coordinates. The reverse process is not so simple. The radius can be computed by the following formula bosed on the Pythagorean Theorem, $x^2+y^2=r^2.$

$T=\sqrt[\phantom{2}]{x^2}+y^2$

If the coordinotes are within quadronts I and $V,$ that is$,x>0,$

$=ta{n}^1(\frac{y}{x})$

Note: The Moth module implements this with the atan function.

The ongle here, $,$ is messured positively in quasrant I and negstively in qusdront IV$.$ The difficulty srises for the other cases. The following table summarizes the possibilities:

$\backslash$table$[[x,y,8],[x0,=0,ta{n}^{-1}(y^{\text{'}}x)-$