Shear Mapping - Elementary Form

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 following applications of shear mapping were noted by William Kingdon Clifford:

"A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area."

"... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle."

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).