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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
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