$
   r=
   \left\{
      (x,y) \in \mathcal{P}
   \right|
   \left.
      ax+by=c
   \right\}
   =
   \left\{
      \overrightarrow{OP}(t,0)
   \left|
   \right.~
     t \in \mathbb{R}
   \right\}
$.