Two pinhole cameras photograph different scenes. When can the resulting images be identical up to a projective transformation? This is a fundamental ill-posedness question in computer vision: if two different point configurations can produce the same image, the reconstruction problem has an ambiguity that no algorithm can resolve.
Ottaviani and Thomas prove: projective equivalence can occur only when both point sets have at most seven elements. Eight or more points, and the images always differ projectively. The bound is sharp.
The proof uses classical invariant theory and moduli space geometry. For each number of points up to seven, they characterize the Zariski closure of the locus of camera centers that produce projectively equivalent images — the “centers-variety.” As the number of points increases, this variety shrinks. At seven points, the geometry involves the Goepel variety, a classical object from the theory of abelian surfaces. At eight points, the variety becomes empty.
The result means that for generic scenes with eight or more distinguishable points, the camera's position and orientation can in principle be uniquely recovered from its image (up to projective ambiguity of the camera itself). Below eight points, there exist genuine geometric ambiguities where different camera positions photograph different point sets yet produce the same image.
Seven points. That's the boundary between recoverable and ambiguous. The number comes from the geometry of projective space, not from engineering constraints or noise models. It's a theorem about the world, not about cameras.