Shear mapping

In this shear mapping of an image of the Mona Lisa, the picture was deformed in such a way that its central vertical axis was not modified.

In mathematics, a shear mapping 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 horizontal shear (or shear parallel to the x axis) is represented by the linear mapping

This leaves horizontal lines y = c invariant, but for m ≠ 0 maps vertical lines x = a into lines y' = (x'  − a)/m having slope 1/m

Substituting 1/m for m in the matrix gives lines y = m(x − a) of slope m, if desired.

A vertical shear (or shear parallel to the y axis) of lines is accomplished by the linear mapping

The vertical shear leaves vertical lines x = a invariant, but maps horizontal lines y = b into lines y' = mx'  + b

The matrices above 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.

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 (see external link).

Advanced form

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 + Mw ′) + w ′

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

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

See also

• Equi-areal mapping

References

• Weisstein, Eric W. "Shear" from Mathworld, A Wolfram Web Resource.
• Mike May S.J. Pythagorean theorem by shear mapping Saint Louis University website; requires Java and Geogebra. Click on the "Steps" slider and observe shears at steps 5 and 6.