  T-curves are piecewise linear curves which have been used with success since
the beginning of the 1990's to construct new real algebraic curves with
prescribed topology mainly on the real projective plane. In fact T-curves can
be used on any real projective toric surface. We generalize here the
construction of the latter by departing from non-convex polygons and we get a
class of surfaces, the ambient surfaces, that we characterize topologically and
on which T-curves are also well defined. Associated to each T-curve we
construct a surface, the T-filling, which is topologically the analog of the
quotient of the complexification of a real algebraic curve by the complex
conjugation. We use then the T-filling to prove a theorem for T-curves on
arbitrary ambient surfaces which is the analog of a theorem of Harnack for
algebraic curves, and to define the orientability and orientation of a T-curve,
the same way it is defined in the real algebraic setting.
