Pohlke's theorem is the fundamental theorem of
axonometry
Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coord ...
. It was established 1853 by the German painter and teacher of
descriptive geometry
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and ...
Karl Wilhelm Pohlke. The first proof of the theorem was published 1864 by the German mathematician
Hermann Amandus Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.
Life
Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kumme ...
, who was a student of Pohlke. Therefore the theorem is sometimes called theorem of Pohlke and Schwarz, too.
The theorem
![Axonom-pohlke](https://upload.wikimedia.org/wikipedia/commons/7/73/Axonom-pohlke.svg)
*Three arbitrary line sections
in a plane originating at point
, which are not contained in a line, can be considered as the
parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or '' image plane'', where the ''rays'', known as '' li ...
of three edges
of a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
.
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
by the axonometric procedure below.
Pohlke's theorem can be stated in terms of linear algebra as:
*Any
affine mapping
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves line (geometry), lines and parallelism (geometry), parallelism, but not necessarily Euclidean ...
of the 3-dimensional space onto a plane can be considered as the composition of a
similarity and a parallel projection.
Application to axonometry
![Axonom-def](https://upload.wikimedia.org/wikipedia/commons/9/9c/Axonom-def.svg)
Pohlke's theorem is the justification for the following easy procedure to construct a scaled parallel projection of a 3-dimensional object using coordinates,:
#Choose the images of the coordinate axes, not contained in a line.
#Choose for any coordinate axis forshortenings
#The image
of a point
is determined by the three steps, starting at point
:
::go
in
-direction, then
::go
in
-direction, then
::go
in
-direction and
:4. mark the point as
.
In order to get undistorted pictures, one has to choose the images of the axes and the forshortenings carefully (see
Axonometry
Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coord ...
). In order to get an
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
only the images of the axes are free and the forshortenings are determined. (see
:de:orthogonale Axonometrie).
Remarks on Schwarz's proof
Schwarz formulated and proved the more general statement:
*The vertices of any
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
can be considered as an oblique parallel projection of the vertices of a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
that is
similar to a given tetrahedron.
and used a theorem of
L’Huilier:
*Every triangle can be considered as the orthographic projection of a triangle of a given shape.
Notes
{{Reflist
References
*
K. Pohlke: ''Zehn Tafeln zur darstellenden Geometrie.'' Gaertner-Verlag, Berlin 187
(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
External links
F. Klein: ''The fundamental Theorem of Pohlke'', in ''Elementary Mathematics from a Higher Standpoint: Volume II: Geometry'', p. 97Christoph J. Scriba,Peter Schreiber: 5000 Years of Geometry: Mathematics in History and Culture, p. 398.''Pohlke–Schwarz theorem'', Encyclopedia of Mathematics.
Graphical projections
Linear algebra