id: 06322209
dt: j
an: 2015e.00608
au: Bergold, Helmut
ti: A generalization of Pohlke’s theorem. (Eine Verallgemeinerung des Satzes
von Pohlke.)
so: Elem. Math. 69, No. 2, 57-60 (2014).
py: 2014
pu: European Mathematical Society (EMS) Publishing House, Zurich
la: DE
cc: G40 H60
ut: Pohlke’s theorem; elementary geometry; orthogonal projections
ci:
li: doi:10.4171/EM/247
ab: 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.
rv: Wolfgang Globke (Adelaide)