# Shear mapping

Last updated Horizontal shearing of the plane, transforming the blue into the red shape. The black dot is the origin. In fluid dynamics a shear mapping depicts fluid flow between parallel plates in relative motion.

In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin.  This type of mapping is also called shear transformation, transvection, or just shearing.

## Contents

An example is the mapping that takes any point with coordinates $(x,y)$ to the point $(x+2y,y)$ . In this case, the displacement is horizontal by a factor of 2 where the fixed line is the $x$ -axis, and the signed distance is the $y$ coordinate. Note that points on opposite sides of the reference line are displaced in opposite directions.

Shear mappings must not be confused with rotations. Applying a shear map to a set of points of the plane will change all angles between them (except straight angles), 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 parallelograms, and circles into ellipses. 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 slanted (or italic) styles of letters.

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 $n$ -dimensional Cartesian space $\mathbb {R} ^{n}$ , the distance is measured from a fixed hyperplane parallel to the direction of displacement. This geometric transformation is a linear transformation of $\mathbb {R} ^{n}$ that preserves the $n$ -dimensional measure (hypervolume) of any set.

## Definition

### Horizontal and vertical shear of the plane Horizontal shear of a square into parallelograms with factors cot⁡(60∘)≈0.58{\displaystyle \cot(60^{\circ })\approx 0.58} and cot⁡(45∘)=1{\displaystyle \cot(45^{\circ })=1}

In the plane $\mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R}$ , a horizontal shear (or shear parallel to the x axis) is a function that takes a generic point with coordinates $(x,y)$ to the point $(x+my,y)$ ; 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 oblique lines with slope $1/m$ . Therefore, the shear factor $m$ is the cotangent of the shear angle$\varphi$ 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 column vector (a 2×1 matrix), the shear mapping can be written as multiplication by a 2×2 matrix:

${\begin{pmatrix}x^{\prime }\\y^{\prime }\end{pmatrix}}={\begin{pmatrix}x+my\\y\end{pmatrix}}={\begin{pmatrix}1&m\\0&1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.$ 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 transposed matrix:

${\begin{pmatrix}x^{\prime }\\y^{\prime }\end{pmatrix}}={\begin{pmatrix}x\\mx+y\end{pmatrix}}={\begin{pmatrix}1&0\\m&1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.$ 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 $\varphi$ to become lines with slope $m$ .

### General shear mappings

For a vector space V and subspace W, a shear fixing W translates all vectors in a direction parallel to W.

To be more precise, if V is the direct sum of W and W, and we write vectors as

v = w + w

correspondingly, the typical shear L fixing W is

L(v) = (Lw + Lw) = (w + Mw) + w,

where M is a linear mapping from W into W. Therefore in block matrix terms L can be represented as

${\begin{pmatrix}I&M\\0&I\end{pmatrix}}.$ ## 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." 

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  as well as the related geometric mean theorem.

An algorithm due to Alan W. Paeth uses 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 pixels at a time. 

In typography, normal text transformed by a shear mapping results in oblique type.

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.

## Related Research Articles A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.

In mathematics, and more specifically in linear algebra, a linear map is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m ; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c". In Euclidean geometry, an affine transformation, or an affinity, is a geometric transformation that preserves lines and parallelism.

In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + , where a and b are real numbers, and ε is a symbol taken to satisfy with . In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides. In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign : a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

In affine geometry, uniform scaling is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group. Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.

In computer graphics, a computer graphics pipeline, rendering pipeline or simply graphics pipeline, is a conceptual model that describes what steps a graphics system needs to perform to render a 3D scene to a 2D screen. Once a 3D model has been created, for instance in a video game or any other 3D computer animation, the graphics pipeline is the process of turning that 3D model into what the computer displays.   Because the steps required for this operation depend on the software and hardware used and the desired display characteristics, there is no universal graphics pipeline suitable for all cases. However, graphics application programming interfaces (APIs) such as Direct3D and OpenGL were created to unify similar steps and to control the graphics pipeline of a given hardware accelerator. These APIs abstract the underlying hardware and keep the programmer away from writing code to manipulate the graphics hardware accelerators.

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.

In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value.

In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures.

The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. When studying these transformations, the dual numbers are often interpreted as representing oriented lines on the plane. The Laguerre transformations map lines to lines, and include in particular all isometries of the plane.

1. Definition according to Weisstein, Eric W. Shear From MathWorld − A Wolfram Web Resource
2. William Kingdon Clifford (1885) Common Sense and the Exact Sciences, page 113
3. Hohenwarter, M Pythagorean theorem by shear mapping; made using GeoGebra. Drag the sliders to observe the shears
4. Alan Paeth (1986), A Fast Algorithm for General Raster Rotation. Proceedings of Graphics Interface '86, pages 77–81.