Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms at a local scale aggregate and bring on large changes in shape, curvature and elongation at a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces. This paper characterizes the parametrization, curvature and metric of smooth surfaces that the eggbox pattern can fit asymptotically, i.e., when the eggbox unit cell parameter becomes infinitely small compared to the typical radius of curvature of the target surface. In particular, it is demonstrated that no finite region of a sphere can be fitted and a systematic method that allows to fit ruled surfaces is presented. As an application, the fitting of a one-sheeted hyperboloid is constructed.