Editing Shear mapping (section)

==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}}