Ambient Surfaces and T-Fillings of T-Curves

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.