  In this paper, we study a family of curves on $S^2$ that defines a
two-dimensional smooth projective plane. We use curve shortening flow to prove
that any two-dimensional smooth projective plane can be smoothly deformed
through a family of smooth projective planes into one which is isomorphic to
the real projective plane. In addition, as a consequence of our main result, we
show that any two smooth embedded curves on $RP^2$ which intersect
transversally at exactly one point converge to two different geodesics under
the flow.
