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\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. 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:  \[  F_{a,b}(d_z) = b(1-\sqrt{d_z/a})_+  \]  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 $0$.