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. 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.

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