Built-in Mathematica Symbol

ShearingMatrix

ShearingMatrix[

, v, n]
gives the matrix corresponding to shearing by

radians along the direction of the vector v, and normal to the vector n.

MORE INFORMATION

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[, {1, 0}, {0, 1}] effectively slants by angle to the right.

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.

EXAMPLES

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Basic Examples (2)

A shearing by

radians along the x axis:

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

A shearing by

radians along the x axis:

In[1]:=

Out[1]//MatrixForm=

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

In[1]:=

Out[1]=

Scope (5)

Shearing along the x axis:

Shearing along the y axis:

Shearing in the x, y plane along the x axis:

Shearing the x, z plane along the x 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

; hence it preserves areas and volumes:

The inverse of ShearingMatrix[

, v, n] is given by ShearingMatrix[-

, v, n]:

The inverse of ShearingMatrix[

, v, n] is also given by ShearingMatrix[

, -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: