- Shear mapping
In
mathematics , a shear or transvection is a particular kind oflinear 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 carryarea s 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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