
%\bigskip
\section{Monte Carlo circle area}

\includegraphics[height=1.5in]{images/mc-circle-square}

% From http://local.wasp.uwa.edu.au/~pbourke/geometry/circlearea/

A circle with radius $r$ and area $\pi/4 r^2$ is bounded by a square of area $2r * 2r = 4r^2$.
Randomly select $n$ points within the square, of which $k$ are within the circle. The ratio of points inside each figure should be the same as the ratio of areas of the two figures. $\frac{k}{n} = \frac{\pi r^2}{4 r^2} = \frac{\pi}{4}$.

\includegraphics[height=2in]{images/mc-circles}

A simple extension can be used to estimate the area of any number of intersecting circles. Select random points within one circle's bounding box, and count how many fall inside every circle. The area of the intersections is then $\frac{k}{n} 4r^2$

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "ref-doc"
%%% End: 
