Pohlke's theorem

Pohlke's theorem

• Three arbitrary line sections ${\displaystyle {\overline {O}}{\overline {U}},{\overline {O}}{\overline {V}},{\overline {O}}{\overline {W}}}$ in a plane originating at point ${\displaystyle {\overline {O}}}$, which are not contained in a line, can be considered as the parallel projection of three edges ${\displaystyle OU,OV,OW}$ of a cube.

For a mapping of a unit cube, one has to apply an additional scaling either in the space or in the plane. Because a parallel projection and a scaling preserves ratios one can map an arbitrary point ${\displaystyle P=(x,y,z)}$ by the axonometric procedure below.

Pohlke's theorem can be stated in terms of linear algebra as:

• Any affine mapping of the 3-dimensional space onto a plane can be considered as the composition of a similarity and a parallel projection.[1]

the principle of axonometric projection

Pohlke's theorem is the justification for the following easy procedure to construct a scaled parallel projection of a 3-dimensional object using coordinates,:[2][3]

1. Choose the images of the coordinate axes, not contained in a line.
2. Choose for any coordinate axis forshortenings ${\displaystyle v_{x},v_{y},v_{z}>0.}$
3. The image ${\displaystyle {\overline {P}}}$ of a point ${\displaystyle P=(x,y,z)}$ is determined by the three steps, starting at point ${\displaystyle {\overline {O}}}$:

go ${\displaystyle v_{x}\cdot x}$ in ${\displaystyle {\overline {x}}}$-direction, then

go ${\displaystyle v_{y}\cdot y}$ in ${\displaystyle {\overline {y}}}$-direction, then

go ${\displaystyle v_{z}\cdot z}$ in ${\displaystyle {\overline {z}}}$-direction and

4. mark the point as ${\displaystyle {\overline {P}}}$.

In order to get undistorted pictures, one has to choose the images of the axes and the forshortenings carefully (see Axonometry). In order to get an orthographic projection only the images of the axes are free and the forshortenings are determined. (see de:orthogonale Axonometrie).

Schwarz formulated and proved the more general statement:

• The vertices of any quadrilateral can be considered as an oblique parallel projection of the vertices of a tetrahedron that is similar to a given tetrahedron.[4]

and used a theorem of L’Huilier:

• Every triangle can be considered as the orthographic projection of a triangle of a given shape.

• K. Pohlke: Zehn Tafeln zur darstellenden Geometrie. Gaertner-Verlag, Berlin 1876 (Google Books.)
• Schwarz, H. A.:Elementarer Beweis des Pohlkeschen Fundamentalsatzes der Axonometrie,J. reine angew. Math. 63, 309–314, 1864.
• Arnold Emch: Proof of Pohlke's Theorem and Its Generalizations by Affinity, American Journal of Mathematics, Vol. 40, No. 4 (Oct., 1918), pp. 366–374

• F. Klein: The fundamental Theorem of Pohlke, in Elementary Mathematics from a Higher Standpoint: Volume II: Geometry, p. 97,
• Christoph J. Scriba,Peter Schreiber: 5000 Years of Geometry: Mathematics in History and Culture, p. 398.
• Pohlke–Schwarz theorem, Encyclopedia of Mathematics.