$
   H^-=
   \left\{
      (x,y) \in \mathcal{P}
   \right|
   \left.
      ax+by>c
   \right\}
   =
   \left\{
      P \in \mathcal{P}
   \left|
   \right.~
      \overrightarrow{OP}(t,s)
      ~s < 0
   \right\}
$.