History   Help on query formulation   A generalization of Pohlke’s theorem. (Eine Verallgemeinerung des Satzes von Pohlke.) (German. English summary)
Elem. Math. 69, No. 2, 57-60 (2014).
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.
Reviewer: Wolfgang Globke (Adelaide)
Classification: G40 H60