# 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: