The main results of this paper concern the classical problem: if two surfaces in the Euclidean space have congruent projections onto any plane, how different can they be? Here we consider the apparent contours of the smooth hypersurfaces as the projection data and formulate some sufficient conditions of coincidence of the shapes of two hypersurfaces, if the shapes of their apparent contours in any two-dimensional plane coincide. We obtain also new results on reconstruction of smooth surfaces from observations of the wave fronts generated by these surfaces.