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\subsection{Physics of sheep}  Here, we get our inspiration from Boids \cite{boids}, and define the following axioms for movement of sheep:  \begin{description} 
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F_{a,b}(d_z) = b\left(1-\sqrt{\frac{d_z}{a}}\right)_+  \]  and where the direction of the force is given by the angle of $z$ relative to the sheep.  Note that the force function is paramaterized by $a$ and $b$, which have a simple interpetation: $a$ is the maximum distance of the force, and $b$ is the magnitude of the force when the distance is $0$. $0$.\footnote{Most of the physical simulation is currently implemented, code can be found on \href{https://github.com/schmit/sheepherding}{github.com/schmit/sheepherding}.}