Definitions

Shear mapping

In mathematics, a shear or transvection is a particular kind of linear mapping. Its effect leaves fixed all points on one axis and other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from the axis. It is notable that shear mappings carry areas into equal areas.

Elementary form

In the plane {(x,y): x,y ∈ R }, a vertical shear (or shear parallel to the x axis) for m ≠ 0 of vertical lines x = a into lines y = (x - a)/m of slope 1/m is represented by the linear mapping
$\left(x,y\right) begin\left\{pmatrix\right\}1 & 0m & 1end\left\{pmatrix\right\} = \left(x+my,y\right).$
One can substitute 1/m for m in the matrix to get lines y = m(x - a) of slope m if desired.

A horizontal shear (or shear parallel to the y axis) of lines y = b into lines y = mx + b is accomplished by the linear mapping

$\left(x,y\right) begin\left\{pmatrix\right\}1 & m 0 & 1end\left\{pmatrix\right\} = \left(x,mx + y\right).$

These are special cases of shear matrices, which allow for generalization to higher dimensions. The shear elements here are either m or 1/m, case depending.

For a vector space V and subspace W, a shear fixing W translates all vectors 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 fixing W is L where

L(v) = (w + w′M) + w′

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

$begin\left\{pmatrix\right\} I & 0 M & I end\left\{pmatrix\right\}$

with blocks on the diagonal I (identity matrix), with M below the diagonal, and 0 above.