Consider Phong shading, where the surface normal for shading purposes is interpolated from the vertex normals. With Phong shading, faces are still flat, but the normals behave as if they were curved.
Suppose we construct the curved surface which is continuous and perpendicular to all the Phong normals. We then express this surface procedurally as a displacement from the flat face. This yields a procedural displacement map, which, if applied, would create a true curved surface. For example, if a sphere were drawn in this way, it would occlude its background as a true circle, not a polygon.
This could be the next step up from Phong shading. The underlying model is still a mesh, but the illusion of a curved surface doesn't break down at the edges. This could make low-poly models look much better.
Somebody must have thought of this years ago.
But I haven't seen it in the usual textbooks. It might be implementable on modern graphics boards that support displacement mapping, so it's worth looking at.