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ShearingMatrix

ShearingMatrix[θ,v,n]

gives the matrix corresponding to shearing by θ radians along the direction of the vector v, and normal to the vector n.

Details

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 2D, ShearingMatrix turns rectangles into parallelograms. ShearingMatrix[θ,{1,0},{0,1}] effectively slants by angle θ to the right.
In 3D, ShearingMatrix does the analog of shearing a deck of cards by angle θ in the direction v, with the cards being oriented so as to have normal vector n.

Examples

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Basic Examples (2)

A shearing by θ radians along the axis:

Apply a 30° shear along the axis to a square:

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:

Properties & Relations (4)

The determinant of a shearing matrix is 1; hence it preserves areas and volumes:

The inverse of ShearingMatrix[θ,v,n] is given by ShearingMatrix[-θ,v,n]:

The inverse of ShearingMatrix[θ,v,n] is also given by ShearingMatrix[θ,-v,n]:

The power of a shearing matrix is again a shearing matrix with the same and :

Possible Issues (3)

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:

Neat Examples (1)

The transformation applied to a sphere:

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