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. 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 ( x , y ) {\displaystyle (x,y)} to the point ( x + 2 y , y ) {\displaystyle (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. For fonts that do not implement true-italics, 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 R n , {\displaystyle \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 R n {\displaystyle \mathbb {R} ^{n}} that preserves the n-dimensional measure (hypervolume) of any set.
Definition
Horizontal and vertical shear of the plane
In the plane R 2 = R × R {\displaystyle \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 ) {\displaystyle (x,y)} to the point ( x + m y , y ) {\displaystyle (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 . {\displaystyle {\tfrac {1}{m}}.} Therefore, the shear factor m is the cotangent of the shear angle φ {\displaystyle \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:
( x ′ y ′ ) = ( x + m y y ) = ( 1 m 0 1 ) ( x y ) . {\displaystyle {\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}}.}
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