Shear (mathematics)

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In this shear transformation of an image of the Mona Lisa, the picture was deformed in such a way that its central vertical axis was not modified. (Note: The corners have been cropped on the right hand picture.)

In mathematics, a shear is a particular kind of linear mapping.

[edit] Elementary form

In the plane {(x,y): x,y ∈R}, the vertical shear of lines y = b into lines y = mx + b is accomplished by the linear mapping

(x,y) = (x,mx + y).

Similarly, a horizontal shear of vertical lines x = a into lines of slope m is represented by the linear mapping

(x,y) = (x+y/m,y).

[edit] 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 + (w′ + M(w′))

where M is a linear mapping from W′ into W. Therefore in block matrix terms L can be represented as 2×2 , with blocks on the diagonal I (identity matrix), with M above the diagonal, and 0 below.

Mathematical shears are also called transvections.