ShearingMatrix

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`ShearingMatrix`

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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

open allclose all

Basic Examples (2)Summary of the most common use cases

A shearing by θ radians along the axis:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-hhjsl2

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

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-g6p9bp

Out[1]=1

Scope (5)Survey of the scope of standard use cases

Shearing along the axis:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-bsjgcs

Shearing along the axis:

In[2]:=2

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https://wolfram.com/xid/0dczko9v3s0i-hrfth

Shearing in the plane along the axis:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-b0ll0q

Shearing the plane along the axis:

In[2]:=2

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https://wolfram.com/xid/0dczko9v3s0i-cgmoz

A shearing by angle in the direction in the line :

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-rx9kx

Transformation applied to a 2D shape:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-g7xt0l

In[2]:=2

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https://wolfram.com/xid/0dczko9v3s0i-crv30x

Out[2]=2

Transformation applied to a 3D shape:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-gc95ov

In[2]:=2

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https://wolfram.com/xid/0dczko9v3s0i-c1y145

Out[2]=2

Applications (2)Sample problems that can be solved with this function

Applying the transformation to a surface:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-egoo10

Out[1]=1

Generate all simple (directions parallel to axes) shearing matrices for dimension n:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-fziujs

All shearings in 2D:

In[2]:=2

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https://wolfram.com/xid/0dczko9v3s0i-b5hbl9

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dczko9v3s0i-bicbny

Out[3]=3

All shearings in 3D:

In[4]:=4

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https://wolfram.com/xid/0dczko9v3s0i-jinfgw

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0dczko9v3s0i-cge62c

Out[5]=5

All shearings in 4D:

In[6]:=6

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https://wolfram.com/xid/0dczko9v3s0i-bcmoyv

Out[6]=6

Properties & Relations (4)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-do8h5e

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-dlcly9

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-ecu4ee

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-b8sfdk

Out[1]=1

Possible Issues (3)Common pitfalls and unexpected behavior

The order in which shearings are applied is significant:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-bawjb1

Here the two different orders do not yield the same matrix:

In[2]:=2

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https://wolfram.com/xid/0dczko9v3s0i-erq30a

Out[2]=2

The transformation is not defined for angles such that :

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-bu0g5k

Out[1]=1

For non-orthogonal vectors, the direction is determined by the projection of the direction vector:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-glanwq

Out[1]=1

Neat Examples (1)Surprising or curious use cases

The transformation applied to a sphere:

In[1]:=1

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https://wolfram.com/xid/0dczko9v3s0i-b41tud

Out[1]=1

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