Theorem of Pohlke

In general, if one projects, in parallel projection, a trihedron orthonormé $(\backslash \; vec\; I,\; \backslash \; vec\; J,\; \backslash \; vec\; K)$ on a plan P, one obtains a triplet of vectors ($(\backslash \; vec\; U,\; \backslash \; vec\; v,\; \backslash \; vec\; W)$ which generate the plan.

Reciprocally there is the theorem of Pohlke (1855): Any triplet of vectors which generate a plan P, is the image, except for a homothety, of a trihedron orthonormé by an oblique projection.

An extension of the theorem of Pohlke

That is to say three convergent lines distinct from the plan of projection. It is easy to build a triangle of which the three heights are the lines carried by the three lines given. If the orthocentre of this triangle is interior with the triangle, there exists a trirectangular trihedron which is projected orthogonally on these three lines.

In particular if has, C and D are three distinct points out of the three convergent lines out of O, then OACD can be regarded as a corner of cube, representing in riding prospect cube OABCDEFG.

Application

See: Problem with one euro Business of logic n°534 - Le Monde, May 29th - June 5th, 2007 Elisabeth Busser and Gilles Cohen Copyright POLE 2007

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