We obviously would like to avoid computing an integral everytime we need a random variable. If we however switch to polar coordinates, we have for \( x \) and \( y \) $$ \begin{equation*} r=\left(x^2+y^2\right)^{1/2} \hspace{1cm} \theta =tan^{-1}\frac{x}{y}, \end{equation*} $$ resulting in $$ \begin{equation*} g(r,\theta)=r\exp{(-r^2/2)}drd\theta, \end{equation*} $$ where the angle \( \theta \) could be given by a uniform distribution in the region \( [0,2\pi] \). Following example 1 above, this implies simply multiplying random numbers \( x\in [0,1] \) by \( 2\pi \).