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# ShearingMatrix

 ShearingMatrix gives the matrix corresponding to shearing by radians along the direction of the vector v, and normal to the vector n.
• ShearingMatrix gives matrices corresponding to shearing with the origin kept fixed.
• ShearingMatrix gives matrices with determinant 1, corresponding to area- or volume-preserving transformations.
• In 3D, ShearingMatrix does the analog of shearing a deck of cards by angle in the direction v, with the cards being oriented so as to have normal vector n.
A shearing by radians along the axis:
Apply a 30° shear along the axis to a square:
A shearing by radians along the axis:
 Out[1]//MatrixForm=

Apply a 30° shear along the axis to a square:
 Out[1]=
 Scope   (5)
Shearing along the axis:
Shearing along the axis:
Shearing in the plane along the axis:
Shearing the plane along the axis:
A shearing by angle in the direction in the line :
Transformation applied to a 2D shape:
Transformation applied to a 3D shape:
 Applications   (2)
Applying the transformation to a surface:
Generate all simple (directions parallel to axes) shearing matrices for dimension n:
All shearings in 2D:
All shearings in 3D:
All shearings in 4D:
The determinant of a shearing matrix is 1; hence it preserves areas and volumes:
The inverse of ShearingMatrix is given by ShearingMatrix:
The inverse of ShearingMatrix is also given by ShearingMatrix:
The power of a shearing matrix is again a shearing matrix with the same and :
The order in which shearings are applied is significant:
Here the two different orders do not yield the same matrix:
The transformation is not defined for angles such that :
For non-orthogonal vectors, the direction is determined by the projection of the direction vector:
The transformation applied to a sphere:
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