Shear mapping

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: (x,y) egin{pmatrix}1 & 0\m & 1end{pmatrix} = (x+my,y).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: (x,y) egin{pmatrix}1 & m\ 0 & 1end{pmatrix} = (x,mx + y).

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.

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

:egin{pmatrix} I & 0 \ M & I end{pmatrix}

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

ee also

*Equi-areal mapping

References

* Weisstein, Eric W. [http://mathworld.wolfram.com/Shear.html "Shear"] from Mathworld, A Wolfram Web Resource.


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