Shear mapping

<h2 id="definition">Definition</h2>
<h3>Horizontal and vertical shear of the plane</h3>

In the plane 
 
 
 
 
 
 R
 
 
 2
 
 
 =
 
 R
 
 ×
 
 R
 
 
 
 {\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }
 
, a horizontal shear (or shear parallel to the x-axis) is a function that takes a generic point with coordinates 
 
 
 
 (
 x
 ,
 y
 )
 
 
 {\displaystyle (x,y)}
 
 to the point 
 
 
 
 (
 x
 +
 m
 y
 ,
 y
 )
 
 
 {\displaystyle (x+my,y)}
 
; where m is a fixed parameter, called the shear factor.
The effect of this mapping is to displace every point horizontally by an amount proportionally to its y-coordinate. Any point above the x-axis is displaced to the right (increasing x) if m > 0, and to the left if m < 0. Points below the x-axis move in the opposite direction, while points on the axis stay fixed.
Straight lines parallel to the x-axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the x-axis. Vertical lines, in particular, become <a href="/facts/Angle/gFUTEQYq">oblique</a> lines with <a href="/facts/Slope/a2L3k0j5">slope</a> 
 
 
 
 
 
 
 1
 m
 
 
 
 .
 
 
 {\displaystyle {\tfrac {1}{m}}.}
 
 Therefore, the shear factor m is the <a href="/facts/Cotangent/czej7ur0">cotangent</a> of the shear angle 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 between the former verticals and the x-axis. In the example on the right the square is tilted by 30°, so the shear angle is 60°.
If the coordinates of a point are written as a <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> (a 2×1 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>), the shear mapping can be written as <a href="/facts/Matrix_product/lYDB6Ro2">multiplication</a> by a 2×2 matrix:

(
            
              
                
                  
                    x
                    
                      ′
                    
                  
                
              
              
                
                  
                    y
                    
                      ′
                    
                  
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  x
                  +
                  m
                  y
                
              
              
                
                  y
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                
                
                  m
                
              
              
                
                  0
                
                
                  1
                
              
            
            )
          
        
        
          
            (
            
              
                
                  x
                
              
              
                
                  y
                
              
            
            )
          
        
        .
      
    
    {\displaystyle {\begin{pmatrix}x^{\prime }\\y^{\prime }\end{pmatrix}}={\begin{pmatrix}x+my\\y\end{pmatrix}}={\begin{pmatrix}1&m\\0&1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.}

A vertical shear (or shear parallel to the y-axis) of lines is similar, except that the roles of x and y are swapped. It corresponds to multiplying the coordinate vector by the <a href="/facts/Transpose_of_a_matrix/8wmsagGS">transposed matrix</a>:

(
            
              
                
                  
                    x
                    
                      ′
                    
                  
                
              
              
                
                  
                    y
                    
                      ′
                    
                  
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  x
                
              
              
                
                  m
                  x
                  +
                  y
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                
                
                  0
                
              
              
                
                  m
                
                
                  1
                
              
            
            )
          
        
        
          
            (
            
              
                
                  x
                
              
              
                
                  y
                
              
            
            )
          
        
        .
      
    
    {\displaystyle {\begin{pmatrix}x^{\prime }\\y^{\prime }\end{pmatrix}}={\begin{pmatrix}x\\mx+y\end{pmatrix}}={\begin{pmatrix}1&0\\m&1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.}

The vertical shear displaces points to the right of the y-axis up or down, depending on the sign of m. It leaves vertical lines invariant, but tilts all other lines about the point where they meet the y-axis. Horizontal lines, in particular, get tilted by the shear angle 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 to become lines with slope m.

<h4>Composition</h4>
Two or more shear transformations can be combined.
If two shear matrices are 
 
 
 
 
 
 (
 
 
 
 1
 
 
 λ
 
 
 
 
 0
 
 
 1
 
 
 
 )
 
 
 
 
 {\textstyle {\begin{pmatrix}1&\lambda \\0&1\end{pmatrix}}}
 
 and 
 
 
 
 
 
 (
 
 
 
 1
 
 
 0
 
 
 
 
 μ
 
 
 1
 
 
 
 )
 
 
 
 
 {\textstyle {\begin{pmatrix}1&0\\\mu &1\end{pmatrix}}}

then their composition matrix is

(
            
              
                
                  1
                
                
                  λ
                
              
              
                
                  0
                
                
                  1
                
              
            
            )
          
        
        
          
            (
            
              
                
                  1
                
                
                  0
                
              
              
                
                  μ
                
                
                  1
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                  +
                  λ
                  μ
                
                
                  λ
                
              
              
                
                  μ
                
                
                  1
                
              
            
            )
          
        
        ,
      
    
    {\displaystyle {\begin{pmatrix}1&\lambda \\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\\mu &1\end{pmatrix}}={\begin{pmatrix}1+\lambda \mu &\lambda \\\mu &1\end{pmatrix}},}

which also has determinant 1, so that area is preserved.
In particular, if 
 
 
 
 λ
 =
 μ
 
 
 {\displaystyle \lambda =\mu }
 
, we have

 
 
 
 
 
 (
 
 
 
 1
 +
 
 λ
 
 2
 
 
 
 
 λ
 
 
 
 
 λ
 
 
 1
 
 
 
 )
 
 
 ,
 
 
 {\displaystyle {\begin{pmatrix}1+\lambda ^{2}&\lambda \\\lambda &1\end{pmatrix}},}

which is a <a href="/facts/Positive_definite_matrix/Hbr6AuPS">positive definite matrix</a>.

<h3>Higher dimensions</h3>
A typical shear matrix is of the form

S
        =
        
          
            (
            
              
                
                  1
                
                
                  0
                
                
                  0
                
                
                  λ
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            )
          
        
        .
      
    
    {\displaystyle S={\begin{pmatrix}1&0&0&\lambda &0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{pmatrix}}.}

This matrix shears parallel to the x axis in the direction of the fourth dimension of the underlying vector space.
A shear parallel to the x axis results in 
 
 
 
 
 x
 ′
 
 =
 x
 +
 λ
 y
 
 
 {\displaystyle x'=x+\lambda y}
 
 and 
 
 
 
 
 y
 ′
 
 =
 y
 
 
 {\displaystyle y'=y}
 
. In matrix form:

(
            
              
                
                  
                    x
                    ′
                  
                
              
              
                
                  
                    y
                    ′
                  
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                
                
                  λ
                
              
              
                
                  0
                
                
                  1
                
              
            
            )
          
        
        
          
            (
            
              
                
                  x
                
              
              
                
                  y
                
              
            
            )
          
        
        .
      
    
    {\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}={\begin{pmatrix}1&\lambda \\0&1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.}

Similarly, a shear parallel to the y axis has 
 
 
 
 
 x
 ′
 
 =
 x
 
 
 {\displaystyle x'=x}
 
 and 
 
 
 
 
 y
 ′
 
 =
 y
 +
 λ
 x
 
 
 {\displaystyle y'=y+\lambda x}
 
. In matrix form:

(
            
              
                
                  
                    x
                    ′
                  
                
              
              
                
                  
                    y
                    ′
                  
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                
                
                  0
                
              
              
                
                  λ
                
                
                  1
                
              
            
            )
          
        
        
          
            (
            
              
                
                  x
                
              
              
                
                  y
                
              
            
            )
          
        
        .
      
    
    {\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}={\begin{pmatrix}1&0\\\lambda &1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.}

In 3D space this matrix shear the YZ plane into the diagonal plane passing through these 3 points: 
 
 
 
 (
 0
 ,
 0
 ,
 0
 )
 
 
 {\displaystyle (0,0,0)}
 
 
 
 
 
 (
 λ
 ,
 1
 ,
 0
 )
 
 
 {\displaystyle (\lambda ,1,0)}
 
 
 
 
 
 (
 μ
 ,
 0
 ,
 1
 )
 
 
 {\displaystyle (\mu ,0,1)}

S
        =
        
          
            (
            
              
                
                  1
                
                
                  λ
                
                
                  μ
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            )
          
        
        .
      
    
    {\displaystyle S={\begin{pmatrix}1&\lambda &\mu \\0&1&0\\0&0&1\end{pmatrix}}.}

The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> will always be 1, as no matter where the shear element is placed, it will be a member of a skew-diagonal that also contains zero elements (as all skew-diagonals have length at least two) hence its product will remain zero and will not contribute to the determinant. Thus every shear matrix has an <a href="/facts/Inverse_matrix/fPqXk3V8">inverse</a>, and the inverse is simply a shear matrix with the shear element negated, representing a shear transformation in the opposite direction. In fact, this is part of an easily derived more general result: if S is a shear matrix with shear element λ, then Sn is a shear matrix whose shear element is simply nλ. Hence, raising a shear matrix to a power n multiplies its shear factor by n.

<h4>Properties</h4>
If S is an n × n shear matrix, then:

<ul><li>S has <a href="/facts/Rank_of_a_matrix/Yp9b5LK3">rank</a> n and therefore is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a></li>
<li>1 is the only <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> of S, so <a href="/facts/Determinant/eGYqJ2TF">det</a> S = 1 and <a href="/facts/Trace_(linear_algebra)/dchFFvHt">tr</a> S = n</li>
<li>the <a href="/facts/Eigenspace/8TjEoT8u">eigenspace</a> of S (associated with the eigenvalue 1) has n − 1 dimensions.</li>
<li>S is <a href="/facts/Defective_matrix/I5uysqAs">defective</a></li>
<li>S is asymmetric</li>
<li>S may be made into a <a href="/facts/Block_matrix/fPI8HX9f">block matrix</a> by at most 1 column interchange and 1 row interchange operation</li>
<li>the <a href="/facts/Area_(geometry)/KsNG1G9t">area</a>, <a href="/facts/Volume_(geometry)/aeAXCBUd">volume</a>, or any higher order interior capacity of a <a href="/facts/Polytope/a76nXrL0">polytope</a> is invariant under the shear transformation of the polytope's vertices.</li></ul>
<h3>General shear mappings</h3>
For a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V and <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> W, a shear fixing W translates all vectors in a direction parallel to W.
To be more precise, if V is the <a href="/facts/Direct_sum_of_vector_spaces/Y6PBJnjg">direct sum</a> of W and W′, and we write vectors as

v
        =
        w
        +
        
          w
          ′
        
      
    
    {\displaystyle v=w+w'}

correspondingly, the typical shear L fixing W is

L
        (
        v
        )
        =
        (
        L
        w
        +
        L
        
          w
          ′
        
        )
        =
        (
        w
        +
        M
        
          w
          ′
        
        )
        +
        
          w
          ′
        
        ,
      
    
    {\displaystyle L(v)=(Lw+Lw')=(w+Mw')+w',}

where M is a linear mapping from W′ into W. Therefore in <a href="/facts/Block_matrix/fPI8HX9f">block matrix</a> terms L can be represented as

(
            
              
                
                  I
                
                
                  M
                
              
              
                
                  0
                
                
                  I
                
              
            
            )
          
        
        .
      
    
    {\displaystyle {\begin{pmatrix}I&M\\0&I\end{pmatrix}}.}

<h2 id="applications">Applications</h2>
The following applications of shear mapping were noted by <a href="/facts/William_Kingdon_Clifford/PKY4D3JI">William Kingdon Clifford</a>:

"A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area."
"... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle."<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
The area-preserving property of a shear mapping can be used for results involving area. For instance, the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a> has been illustrated with shear mapping<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> as well as the related <a href="/facts/Geometric_mean_theorem/BQKf7ya4">geometric mean theorem</a>.
Shear matrices are often used in <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
An algorithm due to <a href="/facts/Alan_W._Paeth/820woKYI">Alan W. Paeth</a> uses <a href="/facts/Rotation_matrix/c2K2Ftl8">a sequence of three shear mappings</a> (horizontal, vertical, then horizontal again) to rotate a <a href="/facts/Digital_image/3TNT47Mv">digital image</a> by an arbitrary angle. The algorithm is very simple to implement, and very efficient, since each step processes only one column or one row of <a href="/facts/Pixel/UzwBvXBV">pixels</a> at a time.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
In <a href="/facts/Typography/byx4sANl">typography</a>, normal text transformed by a shear mapping results in <a href="/facts/Oblique_type/yX303maJ">oblique type</a>.
In pre-Einsteinian <a href="/facts/Galilean_relativity/fFhMN2yv">Galilean relativity</a>, transformations between <a href="/facts/Frames_of_reference/15yYfB9M">frames of reference</a> are shear mappings called <a href="/facts/Galilean_transformations/CR8VWxbr">Galilean transformations</a>. These are also sometimes seen when describing moving reference frames relative to a "preferred" frame, sometimes referred to as <a href="/facts/Absolute_time_and_space/g8sJ1EbD">absolute time and space</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Transformation_matrix/qAwPXmpR">Transformation matrix</a></li></ul>

Wikimedia Commons has media related to Shear (geometry).

The Wikibook Abstract Algebra has a page on the topic of: Shear mapping

<h2 id="bibliography">Bibliography</h2>
<ul><li>Foley, James D.; van Dam, Andries; Feiner, Steven K.; Hughes, John F. (1991), <a href="https://archive.org/details/computergraphics00fole">Computer Graphics: Principles and Practice</a> (2nd ed.), Reading: <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-12110-7</li></ul>

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

<ol>
<li id="fn:1">Weisstein, Eric W. "Shear". MathWorld − A Wolfram Web Resource. Definition according to Weisstein. <a href="http://mathworld.wolfram.com/Shear.html" target="_blank">http://mathworld.wolfram.com/Shear.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Clifford, William Kingdon (1885). Common Sense and the Exact Sciences. p. 113. <a href="/wiki/William_Kingdon_Clifford" target="_blank">/wiki/William_Kingdon_Clifford</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hohenwarter, M. "Pythagorean theorem by shear mapping". Made using GeoGebra. Drag the sliders to observe the shears. <a href="http://tube.geogebra.org/m/125392" target="_blank">http://tube.geogebra.org/m/125392</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Foley et al. (1991, pp. 207–208, 216–217) - Foley, James D.; van Dam, Andries; Feiner, Steven K.; Hughes, John F. (1991), Computer Graphics: Principles and Practice (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-12110-7 <a href="https://archive.org/details/computergraphics00fole" target="_blank">https://archive.org/details/computergraphics00fole</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Schneider, Philip J.; Eberly, David H. (2002). "Geometric Tools for Computer Graphics". pp. 154–157. <a href="https://books.google.com/books?id=3Q7HGBx1uLIC" target="_blank">https://books.google.com/books?id=3Q7HGBx1uLIC</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Desai, Apueva A. "Computer Graphics". pp. 162–164. <a href="https://books.google.com/books?id=WQiIj8ZS0IoC" target="_blank">https://books.google.com/books?id=WQiIj8ZS0IoC</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Paeth, A.W. (1986). "A Fast Algorithm for General Raster Rotation" (PDF). Vision Interface (VI1986). pp. 077–081. <a href="https://www.cipprs.org/archive/vi/VI1986/pp077-081-Paeth-1986.pdf" target="_blank">https://www.cipprs.org/archive/vi/VI1986/pp077-081-Paeth-1986.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Shear mapping open-in-new

Shear mapping