Shear is the translation along an axis (say, X axis) by an amount that increases linearly with another axis (Y). It produces shape distortions as if objects were composed of layers that are caused to slide over each other .
Shear transformations are very useful in creating italic letters and slanted letters from regular letters.
where h is the negative or positive fraction of Y coordinate of P to be added to the X coordinate. can be any real number.
The matrix of form of shear in x-direction is given by
Combining the shear in X and Y directions,
where g is a non-zero fraction of to be added to the Y coordinate.
The general matrix form of shear is
Shear will reduce to a pure shear in the y-direction, when h=0.
The inverse of Shear is given by
Shear in Y direction is
Shear along any pair of axes is proportional to the third axis. For instance, to shear along z in 3D, x and y values are altered by an amount proportional to the value of z, leaving z unchanged. Let , is the shear due to z along x and y directions respectively and are real values. Then the matrix representation is
Shear for x , y axis is similar to that of z. The general form of shear is given by