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Sven Schmit deleted file physical model1.tex
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\section{A physical model}
Underlying all simulations is a physical model that describes the world, and the movement of sheep in particular.
\subsection{The world}
The world is modeled as a box, $[0, x_{m}]\times[0,y_{m}]$.
We can represent the sheep, dogs and target(s) as points.
We choose not to discretize the space, since that allows for more prolific movement, a more realistic environment, and a more interesting modeling perspective.
\subsection{Physics of sheep}
Here, we get our inspiration from Boids \cite{boids}, and define the following axioms for movement of sheep:
\begin{description}
\item[Axiom 1] Sheep move away from dogs when sufficiently close.
\item[Axion 2] When two sheep are very close to each other, they want more space and move away.
\item[Axiom 3] When two sheep are neither too close, nor too far, they flock together.
\end{description}
We translate these axioms into forces that are acting on the sheep. These forces then drive the sheep in a certain direction and define the speed at which the sheep moves.
