First the power cosine distribution as introduced in (2.6) is considered, where with . Thus the probability density function for the polar angle is given as

Calculating the cumulative distribution function results in

(5.37) |

A variate obeying an arbitrary distribution can be obtained using the inversion method [25]. A uniformly distributed variate on mapped by the inverse cumulative distribution function leads to the desired distribution [115]

Since is uniformly distributed on as well, the random polar angle can also be generated by [66]

(5.39) |

Algorithm 5.2 summarizes the generation of a random polar angle following the probability density function (5.36).

Otmar Ertl: Numerical Methods for Topography Simulation