Editing Shear mapping (section)

{{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.