# ShearingTransform

ShearingTransform[θ,v,n]

gives a TransformationFunction that represents a shear by θ radians along the direction of the vector v, normal to the vector n, and keeping the origin fixed.

ShearingTransform[θ,v,n,p]

gives a shear that keeps the point p fixed, rather than the origin.

# Details

• ShearingTransform gives a TransformationFunction which can be applied to vectors.
• ShearingTransform works in any number of dimensions, and always gives area- or volume-preserving transformations.
• In 2D, ShearingTransform turns rectangles into parallelograms. ShearingTransform[θ,{1,0},{0,1}] effectively represents slanting by angle θ to the right.
• In 3D, ShearingTransform does the analog of shearing a deck of cards by angle θ in the direction v, with the cards oriented so as to have normal vector n, and the card that goes through the point p kept fixed.

# Examples

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## Basic Examples(3)

Shearing by θ radians along the axis:

Apply a 30° shearing along the axis to the unit rectangle:

Apply a shearing transform in the plane:

## Scope(5)

Simple shearing along the axis:

Simple shearing along the axis in the plane:

Shearing along the axis in the plane :

Points in the shearing plane are not changed:

Points outside the shearing plane are moved in the shearing direction:

Transformation applied to a 2D shape:

Transformation applied to a 3D shape:

## Applications(2)

Transforming the output of Plot:

Construct a slanted font from an upright font by shearing:

## Properties & Relations(3)

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

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

Performing the shearing transform multiple times corresponds to a single shearing transform:

## Possible Issues(3)

The order in which shearings are applied is significant:

Applying the two shearings in different orders is not equivalent:

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)

Shear a 3D object about a point p:

In the plane:

In the plane:

In the plane:

Wolfram Research (2007), ShearingTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ShearingTransform.html.

#### Text

Wolfram Research (2007), ShearingTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ShearingTransform.html.

#### CMS

Wolfram Language. 2007. "ShearingTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ShearingTransform.html.

#### APA

Wolfram Language. (2007). ShearingTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ShearingTransform.html

#### BibTeX

@misc{reference.wolfram_2024_shearingtransform, author="Wolfram Research", title="{ShearingTransform}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ShearingTransform.html}", note=[Accessed: 25-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_shearingtransform, organization={Wolfram Research}, title={ShearingTransform}, year={2007}, url={https://reference.wolfram.com/language/ref/ShearingTransform.html}, note=[Accessed: 25-June-2024 ]}