  We classify real families of minimal degree rational curves that cover an
embedded rational surface. A corollary is that if the projective closure of a
smooth surface is not biregular isomorphic to the projective closure of the
unit-sphere, then the set of minimal degree rational curves that cover the
surface is either empty or of dimension at most two. Moreover, if these curves
are of minimal degree over the real numbers, but not over the complex numbers,
then almost all the curves are smooth. Our methods lead to an algorithm that
takes as input a real surface parametrization and outputs all real families of
rational curves of lowest possible degree that cover the image surface.

Algebraic Geometry;Metric Geometry