Pohlke's theorem

<h2 id="the-theorem">The theorem</h2>

<ul><li>Three arbitrary line sections 
  
    
      
        
          
            O
            ¯
          
        
        
          
            U
            ¯
          
        
        ,
        
          
            O
            ¯
          
        
        
          
            V
            ¯
          
        
        ,
        
          
            O
            ¯
          
        
        
          
            W
            ¯
          
        
      
    
    {\displaystyle {\overline {O}}{\overline {U}},{\overline {O}}{\overline {V}},{\overline {O}}{\overline {W}}}
  
 in a plane originating at point 
  
    
      
        
          
            O
            ¯
          
        
      
    
    {\displaystyle {\overline {O}}}
  
, which are not contained in a line, can be considered as the <a href="/facts/Parallel_projection/QhSERFh6">parallel projection</a> of three edges 
  
    
      
        O
        U
        ,
        O
        V
        ,
        O
        W
      
    
    {\displaystyle OU,OV,OW}
  
 of a <a href="/facts/Cube/65lUaAqN">cube</a>.</li></ul>
<p>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 
  
    
      
        P
        =
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle P=(x,y,z)}
  
 by the axonometric procedure below.
</p><p>Pohlke's theorem can be stated in terms of linear algebra as:
</p>
<ul><li>Any <a href="/facts/Affine_transformation/BD7yDr10">affine mapping</a> of the 3-dimensional space onto a plane can be considered as the composition of a <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similarity</a> and a parallel projection.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul>
<h2 id="application-to-axonometry">Application to axonometry</h2>

<p>Pohlke's theorem is the justification for the following easy procedure to construct a scaled parallel projection of a 3-dimensional object using coordinates,:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<ol><li>Choose the images of the coordinate axes, not contained in a line.</li>
<li>Choose for any coordinate axis forshortenings 
  
    
      
        
          v
          
            x
          
        
        ,
        
          v
          
            y
          
        
        ,
        
          v
          
            z
          
        
        >
        0.
      
    
    {\displaystyle v_{x},v_{y},v_{z}>0.}
  
</li>
<li>The image 
  
    
      
        
          
            P
            ¯
          
        
      
    
    {\displaystyle {\overline {P}}}
  
 of a point 
  
    
      
        P
        =
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle P=(x,y,z)}
  
 is determined by the three steps, starting at point 
  
    
      
        
          
            O
            ¯
          
        
      
    
    {\displaystyle {\overline {O}}}
  
:</li></ol>
go 
  
    
      
        
          v
          
            x
          
        
        ⋅
        x
      
    
    {\displaystyle v_{x}\cdot x}
  
 in 
  
    
      
        
          
            x
            ¯
          
        
      
    
    {\displaystyle {\overline {x}}}
  
-direction, then
go 
  
    
      
        
          v
          
            y
          
        
        ⋅
        y
      
    
    {\displaystyle v_{y}\cdot y}
  
 in 
  
    
      
        
          
            y
            ¯
          
        
      
    
    {\displaystyle {\overline {y}}}
  
-direction, then
go 
  
    
      
        
          v
          
            z
          
        
        ⋅
        z
      
    
    {\displaystyle v_{z}\cdot z}
  
 in 
  
    
      
        
          
            z
            ¯
          
        
      
    
    {\displaystyle {\overline {z}}}
  
-direction and
4. mark the point as 
  
    
      
        
          
            P
            ¯
          
        
      
    
    {\displaystyle {\overline {P}}}
  
.
<p>In order to get undistorted pictures, one has to choose the images of the axes and the forshortenings carefully (see <a href="/facts/Axonometry/wR2bk2sN">Axonometry</a>). In order to get an <a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projection</a> only the images of the axes are free and the forshortenings are determined. (see de:orthogonale Axonometrie).
</p>
<h2 id="remarks-on-schwarzs-proof">Remarks on Schwarz's proof</h2>
<p>Schwarz formulated and proved the more general statement:
</p>
<ul><li>The vertices of any <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> can be considered as an oblique parallel projection of the vertices of a <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> that is <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similar</a> to a given tetrahedron.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul>
<p>and used a theorem of <a href="/facts/Lhuilier/ntqAdULl">L’Huilier</a>: 
</p>
<ul><li>Every triangle can be considered as the orthographic projection of a triangle of a given shape.</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Karl_Pohlke/GAxszfIU">K. Pohlke</a>: <i>Zehn Tafeln zur darstellenden Geometrie.</i> Gaertner-Verlag, Berlin 1876 <a href="https://books.google.com/books?id=V17kAAAAMAAJ&q=Karl+Pohlke">(Google Books.)</a></li>
<li><a href="/facts/Hermann_Schwarz/lP1VRdjH">Schwarz, H. A.</a>:<i>Elementarer Beweis des Pohlkeschen Fundamentalsatzes der Axonometrie</i>, J. reine angew. Math. 63, 309–314, 1864.</li>
<li>Arnold Emch: <i>Proof of Pohlke's Theorem and Its Generalizations by Affinity</i>, American Journal of Mathematics, Vol. 40, No. 4 (Oct., 1918), pp. 366–374</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://books.google.com/books?id=3IqQDAAAQBAJ&dq=pohlke%27s+theorem&pg=PA97">F. Klein: <i>The fundamental Theorem of Pohlke</i>, in <i>Elementary Mathematics from a Higher Standpoint: Volume II: Geometry</i>, p. 97</a>,</li>
<li><a href="https://books.google.com/books?id=6Kp9CAAAQBAJ&dq=pohlke%27s+theorem&pg=PA398">Christoph J. Scriba, Peter Schreiber: 5000 Years of Geometry: Mathematics in History and Culture, p. 398.</a></li>
<li><a href="https://www.encyclopediaofmath.org/index.php/Pohlke-Schwarz_theorem"><i>Pohlke–Schwarz theorem</i>, Encyclopedia of Mathematics. </a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>G. Pickert: Vom Satz von Pohlke zur linearen Algebra, Didaktik der Mathematik 11 (1983), 4, pp. 297–306. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Ulrich Graf, Martin Barner: Darstellende Geometrie. Quelle & Meyer, Heidelberg 1961, ISBN 3-494-00488-9, p.144. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Roland Stärk: Darstellende Geometrie, Schöningh, 1978, ISBN 3-506-37443-5, p.156. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Sklenáriková, Zita; Pémová, Marta (2007). "The Pohlke–Schwarz Theorem and its Relevancy in the Didactics of Mathematics" (PDF). Quaderni di Ricerca in Didattica (17). G.R.I.M. (Department of Mathematics, University of Palermo, Italy): 155. <a href="http://math.unipa.it/~grim/quad17_sklenarikova-pemova_07.pdf" target="_blank">http://math.unipa.it/~grim/quad17_sklenarikova-pemova_07.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Pohlke's theorem open-in-new

Pohlke's theorem