A generalization of Pohlke's theorem (Q2509487)

This generalised theorem states: Any two parallelepipeds in 3-space can be rotated in such a way that they ``look identical'', meaning that their projections onto a plane perpendicular to the direction of view are congruent. For the proof, one can assume that the orthogonal projection \(O\) is along the 3-axis onto the 1-2-plane. The vectors spanning the edges of the two parallelepipeds are written as the columns of the two matrices \(V_1\) and \(V_2\), respectively. The statement of the theorem then translates into \(OW_1 V_1=OW_2 V_2\) for suitable conformal-orthogonal transformation matrices \(W_1,W_2\). The author then proves that for any 3-by-3-matrix \(M\) of rank \(\geq 2\), there exist suitable \(W_1,W_2\) such that \(OW_1M=OW_2\). In particular, this holds for \(M=V_1 V_2^{-1}\), which proves the theorem.