deletions | additions       diff --git a/physical model.tex b/physical model.tex  index 5590f80..a4cf452 100644  --- a/physical model.tex  +++ b/physical model.tex 
...
\subsection{The world}  The world is modeled as a box, $[0, x_{m}]\times[0,y_{m}]$.   We can then  represent the sheep, dogs and target(s) as points.   We points in this space.  The majority of the literature uses a grid to model the world, but we  choose not to discretize the space, since that allows for more prolific movement, a more realistic environment, and a more interesting modeling perspective. Sometimes, people add other agents or obstacles to the world, but we propose to leave those out to start with.  \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 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. We note that others use similar approaches.  All forces are calculated using the distance between the sheep and the other object, $z$, denoted by $d_z$, and are of the following form:  \[