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5.3.1 Power Cosine Distribution

First the power cosine distribution as introduced in (2.6) is considered, where $ {g}({\theta})=\left(\cos{\theta}\right)^{\nu}$ with $ {\nu}>0$ . Thus the probability density function for the polar angle is given as

$\displaystyle {f}_{\theta}({\theta})=\left(\cos{\theta}\right)^{\nu}\sin{\theta}.$ (5.36)

Calculating the cumulative distribution function results in

$\displaystyle {F}_{\theta}({\theta}) = \frac{\int_{0}^{{\theta}}{f}_{\theta}({\... ...{f}_{\theta}({\theta}') \,{d}{\theta}} = 1-\left(\cos{\theta}\right)^{{\nu}+1}.$ (5.37)

A variate obeying an arbitrary distribution can be obtained using the inversion method [25]. A uniformly distributed variate $ {u}$ on $ \left[0,1\right]$ mapped by the inverse cumulative distribution function leads to the desired distribution [115]

$\displaystyle {\theta}={F}_{\theta}^{-1}({u})=\arccos\left(\sqrt[{\nu}+1]{1-{u}}\right).$ (5.38)

Since $ 1-{u}$ is uniformly distributed on $ \left[0,1\right]$ as well, the random polar angle can also be generated by [66]

$\displaystyle {\theta}=\arccos\left(\sqrt[{\nu}+1]{{u}}\right).$ (5.39)

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


\begin{algorithm} % latex2html id marker 9424\caption{Generation of a $\left(\... ...turn $\sqrt[{\nu}+1]{\funcrand ()}$\ instead. \end{algorithmic}} \end{algorithm}

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Next: 5.3.2 Coned Cosine Distribution Up: 5.3 Generation of Random Previous: 5.3 Generation of Random
Otmar Ertl: Numerical Methods for Topography Simulation