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 an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance from a given line parallel to that direction.[1] 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 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.
An example is the linear map that takes any point with coordinates to the point . 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 the distance is measured from a fixed hyperplane parallel to the direction of displacement. This geometric transformation is a linear transformation of that preserves the n-dimensional measure (hypervolume) of any set.
^Definition according to Weisstein, Eric W. Shear From MathWorld − A Wolfram Web Resource
In plane geometry, a shearmapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed...
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pp. 3–4. ISBN 978-0486134789. See for example Pythagorean theorem by shearmapping Archived 2016-10-14 at the Wayback Machine, Saint Louis University website...
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dimension Examples of vector spaces Linear map Shearmapping or Galilean transformation Squeeze mapping or Lorentz transformation Linear subspace Row and...
j\\1&k=l\\m&k=i,l=j\end{cases}}} These transformations are a kind of shearmapping, also known as a transvections. The inverse of this matrix is given...
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another one. Case 4 corresponds to a shearmapping combined with a dilation. Case 5 corresponds to a shearmapping combined with a dilation. Case 6 corresponds...
of a rotation in the dual number plane is equivalent to a vertical shearmapping since (1 + pε)(1 + qε) = 1 + (p + q)ε. In absolute space and time the...
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the language of linear algebra, this transformation is considered a shearmapping, and is described with a matrix acting on a vector. With motion parallel...
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