Editing Shear mapping

{{Short description|Type of geometric transformation}}
{{more footnotes needed|date=April 2025}}
[[File:VerticalShear m=1.25 (blue and red).svg|thumb|250px|right|alt=Mesh Shear 5/4|Horizontal shearing of the plane, transforming the blue into the red shape. The black dot is the origin.]]
[[File:Laminar_shear.svg|thumb|250px|right|In [[fluid dynamics]] a shear mapping depicts fluid flow between parallel plates in relative motion.]]

In [[plane geometry]], a '''shear mapping''' is an [[affine transformation]] that displaces each point in a fixed direction by an amount proportional to its [[signed distance function|signed distance]] from a given [[straight line|line]] [[parallel (geometry)|parallel]] to that direction.<ref>{{cite web |quote=Definition according to Weisstein. |last=Weisstein |first=Eric W. |url=http://mathworld.wolfram.com/Shear.html |title=Shear |work=MathWorld − A Wolfram Web Resource }}</ref>

This type of mapping is also called '''shear transformation''', '''transvection''', or just '''shearing'''. The transformations can be applied with a '''shear matrix''' or '''transvection''', an [[elementary matrix]] that represents the [[Elementary row operations#Row-addition transformations|addition]] of a multiple of one row or column to another. Such a [[matrix (mathematics)|matrix]] may be derived by taking the [[identity matrix]] and replacing one of the zero elements with a non-zero value.

An example is the [[linear map]] that takes any point with [[Cartesian coordinates|coordinates]] <math>(x,y)</math> to the point <math>(x + 2y,y)</math>.  In this case, the displacement is horizontal by a factor of 2 where the fixed line is the {{mvar|x}}-axis, and the signed distance is the {{mvar|y}}-coordinate.  Note that points on opposite sides of the reference line are displaced in opposite directions.

Shear mappings must not be confused with [[rotation (geometry)|rotation]]s. Applying a shear map to a set of points of the plane will change all [[angle]]s between them (except [[straight angle]]s), and the length of any [[line segment]] that is not parallel to the direction of displacement. Therefore, it will usually distort the shape of a geometric figure, for example turning squares into [[parallelogram]]s, and [[circle]]s into [[ellipse]]s.  However a shearing does preserve the [[area]] of geometric figures and the alignment and relative distances of [[collinear]] points.  A shear mapping is the main difference between the upright and [[italic font|slanted (or italic)]] styles of [[Latin alphabet|letter]]s.

The same definition is used in [[three-dimensional geometry]], except that the distance is measured from a fixed plane.  A three-dimensional shearing transformation preserves the volume of solid figures, but changes areas of plane figures (except those that are parallel to the displacement). 
This transformation is used to describe [[laminar flow]] of a fluid between plates, one moving in a plane above and parallel to the first.

In the general {{mvar|n}}-dimensional [[Cartesian geometry|Cartesian space]] {{tmath|\R^n,}} the distance is measured from a fixed [[hyperplane]] parallel to the direction of displacement.  This geometric transformation is a [[linear transformation]] of {{tmath|\R^n}} that preserves the {{mvar|n}}-dimensional [[measure (mathematics)|measure]] (hypervolume) of any set.

==Definition==

===Horizontal and vertical shear of the plane===
{{refimprove section|date=April 2025}}
[[File:SVG skewX.svg|thumb|250px|Horizontal shear of a square into parallelograms with factors <math>\cot(60^\circ) = \tan(30^\circ) \approx 0.58</math> and <math>\cot(45^\circ) = \tan(45^\circ) = 1</math>]]
In the plane <math>\R^2 = \R\times\R</math>, a '''horizontal shear''' (or '''shear parallel''' to the {{mvar|x}}-axis) is a function that takes a generic point with coordinates <math>(x,y)</math> to the point <math>(x + m y,y)</math>; where {{mvar|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 {{mvar|y}}-coordinate.  Any point above the {{mvar|x}}-axis is displaced to the right (increasing {{mvar|x}}) if {{math|''m'' > 0}}, and to the left if {{math|''m'' < 0}}. Points below the {{mvar|x}}-axis move in the opposite direction, while points on the axis stay fixed.

Straight lines parallel to the {{mvar|x}}-axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the {{mvar|x}}-axis. Vertical lines, in particular, become [[Angle#Types of angles|oblique]] lines with [[slope]] <math>\tfrac 1 m.</math> Therefore, the shear factor {{mvar|m}} is the [[cotangent]] of the '''shear angle''' <math>\varphi</math> between the former verticals and the {{mvar|x}}-axis.{{fact|date=April 2025}} 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 [[column vector]] (a 2×1 [[matrix (mathematics)|matrix]]), the shear mapping can be written as [[matrix product|multiplication]] by a 2×2 matrix:
: <math>
  \begin{pmatrix}x^\prime \\y^\prime \end{pmatrix}  =
  \begin{pmatrix}x + m y \\y \end{pmatrix} =
  \begin{pmatrix}1 & m\\0 & 1\end{pmatrix} 
    \begin{pmatrix}x \\y \end{pmatrix}.
</math>

A '''vertical shear''' (or shear parallel to the {{mvar|y}}-axis) of lines is similar, except that the roles of {{mvar|x}} and {{mvar|y}} are swapped. It corresponds to multiplying the coordinate vector by the [[transpose of a matrix|transposed matrix]]:

:<math>
  \begin{pmatrix}x^\prime \\y^\prime \end{pmatrix}  = 
  \begin{pmatrix}x \\ m x + y \end{pmatrix} = 
  \begin{pmatrix}1 & 0\\m & 1\end{pmatrix} 
    \begin{pmatrix}x \\y \end{pmatrix}.
</math>

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

====Composition====
Two or more shear transformations can be combined.

If two shear matrices are <math display="inline">\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}</math> and <math display="inline">\begin{pmatrix} 1 & 0 \\ \mu & 1 \end{pmatrix}</math>

then their composition matrix is
<math display="block">\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},</math>
which also has determinant 1, so that area is preserved.

In particular, if <math>\lambda=\mu</math>, we have

<math display="block">\begin{pmatrix} 1 + \lambda^2 & \lambda \\ \lambda & 1 \end{pmatrix},</math>

which is a [[positive definite matrix]].

===Higher dimensions===
A typical shear matrix is of the form
<math display="block">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}.</math>

This matrix shears parallel to the {{mvar|x}} axis in the direction of the fourth dimension of the underlying vector space.

A shear parallel to the {{mvar|x}} axis results in <math>x' = x + \lambda y</math> and <math>y' = y</math>. In matrix form:
<math display="block">\begin{pmatrix} x' \\ y' \end{pmatrix} = 
\begin{pmatrix}
1 & \lambda \\
0 & 1
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}.</math>

Similarly, a shear parallel to the {{mvar|y}} axis has <math>x' = x</math> and <math>y' = y + \lambda x</math>. In matrix form:
<math display="block">\begin{pmatrix}x' \\ y' \end{pmatrix} =
\begin{pmatrix}
1 & 0 \\
\lambda & 1
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}.</math>

In 3D space this matrix shear the YZ plane into the diagonal plane passing through these 3 points: <math>(0, 0, 0)</math> <math>(\lambda, 1, 0)</math> <math>(\mu, 0, 1)</math> 
<math display="block">S = \begin{pmatrix}
1 & \lambda & \mu \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}.</math>

The [[determinant]] 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 [[inverse matrix|inverse]], 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 {{mvar|S}} is a shear matrix with shear element {{math|&lambda;}}, then {{mvar|S<sup>n</sup>}} is a shear matrix whose shear element is simply {{math|''n''&lambda;}}. Hence, raising a shear matrix to a power {{mvar|n}} multiplies its [[shear mapping#Definition|shear factor]] by {{mvar|n}}.

====Properties====
If {{mvar|S}} is an {{math|''n'' × ''n''}} shear matrix, then:
* {{mvar|S}} has [[rank of a matrix|rank]] {{mvar|n}} and therefore is [[invertible matrix|invertible]]
* 1 is the only [[eigenvalue]] of {{mvar|S}}, so {{math|1=[[determinant|det]] ''S'' = 1}} and {{math|1=[[Trace (linear algebra)|tr]] ''S'' = ''n''}}
* the [[eigenspace]] of {{mvar|S}} (associated with the eigenvalue 1) has {{math|''n'' − 1}} dimensions.
* {{mvar|S}} is [[Defective matrix|defective]]
* {{mvar|S}} is asymmetric
* {{mvar|S}} may be made into a [[block matrix]] by at most 1 column interchange and 1 row interchange operation
* the [[area (geometry)|area]], [[volume (geometry)|volume]], or any higher order interior capacity of a [[polytope]] is invariant under the shear transformation of the polytope's vertices.

===General shear mappings===
For a [[vector space]] {{mvar|V}} and [[Linear subspace|subspace]] {{mvar|W}}, a shear fixing {{mvar|W}} translates all vectors in a direction parallel to {{mvar|W}}.

To be more precise, if {{mvar|V}} is the [[direct sum of vector spaces|direct sum]] of {{mvar|W}} and {{mvar|W&prime;}}, and we write vectors as

:<math>v=w+w'</math>

correspondingly, the typical shear {{mvar|L}} fixing {{mvar|W}} is
:<math>L(v) = (Lw+Lw') = (w+Mw') + w',</math>
where {{mvar|M}} is a linear mapping from {{mvar|W&prime;}} into {{mvar|W}}. Therefore in [[block matrix]] terms {{mvar|L}} can be represented as

:<math>\begin{pmatrix} I & M \\ 0 & I \end{pmatrix}. </math>


==Applications==
The following applications of shear mapping were noted by [[William Kingdon Clifford]]:
:"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."<ref>{{cite book |author-link=William Kingdon Clifford |first=William Kingdon |last=Clifford |date=1885 |title=Common Sense and the Exact Sciences |page=113 }}</ref>

The area-preserving property of a shear mapping can be used for results involving area. For instance, the [[Pythagorean theorem]] has been illustrated with shear mapping<ref>{{cite web |last=Hohenwarter |first=M |url=http://tube.geogebra.org/m/125392 |title=Pythagorean theorem by shear mapping |quote=Made using [[GeoGebra]]. Drag the sliders to observe the shears. }}</ref> as well as the related [[Geometric_mean_theorem#Based_on_shear_mappings|geometric mean theorem]].

Shear matrices are often used in [[computer graphics]].<ref>{{harvtxt|Foley|van Dam|Feiner|Hughes|1991|pp=207–208,216–217}}</ref><ref>{{cite book |url=https://books.google.com/books?id=3Q7HGBx1uLIC |title=Geometric Tools for Computer Graphics |first1=Philip J. |last1=Schneider |first2=David H. |last2=Eberly |pages=154–157 |date=2002 |publisher=Elsevier |isbn=978-0-08-047802-9 }}</ref><ref>{{cite book |url=https://books.google.com/books?id=WQiIj8ZS0IoC |title=Computer Graphics |first=Apueva A. |last=Desai |date=22 October 2008 |pages=162–164 |publisher=PHI Learning Pvt. |isbn=978-81-203-3524-0 }}</ref>

An algorithm due to [[Alan W. Paeth]] uses [[Rotation_matrix#Decomposition_into_shears|a sequence of three shear mappings]] (horizontal, vertical, then horizontal again) to rotate a [[digital image]] 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 [[pixel]]s at a time.<ref>{{cite web |first=A.W. |last=Paeth |date=1986 |url=https://www.cipprs.org/archive/vi/VI1986/pp077-081-Paeth-1986.pdf |title=A Fast Algorithm for General Raster Rotation |work=Vision Interface (VI1986) |pages=077–081 }}</ref>

In [[typography]], normal text transformed by a shear mapping results in [[oblique type]].{{fact|date=April 2025}}

In pre-Einsteinian [[Galilean relativity]], transformations between [[frames of reference]] are shear mappings called [[Galilean transformations]]. These are also sometimes seen when describing moving reference frames relative to a "preferred" frame, sometimes referred to as [[absolute time and space]].{{fact|date=April 2025}}

== Etymology ==
The term 'shear' originates from [[Physics]], used to describe a ''cutting-like'' deformation in which parallel layers of material 'slide past each other'. More formally, [[Shear stress|shear force]] refers to unaligned [[Force|forces]] acting on one part of a [[Rigid body|body]] in a specific direction, and another part of the body in the opposite direction.

==See also==
* [[Transformation matrix]]

==References==
{{commons category|Shear (geometry)}}
{{Wikibooks |Abstract Algebra|Shear and Slope|Shear mapping}}
{{reflist|1}}

==Bibliography==
* {{citation | first1 = James D. | last1 = Foley | first2 = Andries | last2 = van Dam | first3 = Steven K. | last3 = Feiner | first4 = John F. | last4 = Hughes | year = 1991 | isbn = 0-201-12110-7 | title = Computer Graphics: Principles and Practice | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading | url-access = registration | url = https://archive.org/details/computergraphics00fole }}

{{Computer graphics}}
{{Matrix classes}}

[[Category:Functions and mappings]]
[[Category:Linear algebra]]