Shearing Objects with a Transformation Matrix

Shearing is a transformation that skews the coordinate space. The idea is to add a multiple of one coordinate to another.

Published on Fri, Feb 5, 2016
Last modified on Mon, Sep 8, 2025
353 words - Page Source

This article is part 3 in the series about transformation matrices:

2D Shearing

In 2D, we can skew points towards the $x$ -axis by making $x^{'} = x + s y$ . If $s > 0$ , then points will skew towards the positive $x$ -axis; if $s < 0$ , points will move towards the negative $x$ -axis.

The transformation matrix that skews points towards the $x$ -axis is:

\begin{matrix} (1) & H_{x} (s) = [\begin{matrix} 1 & s \\ 0 & 1 \end{matrix}] \end{matrix}

Towards the $y$ -axis is:

\begin{matrix} (2) & H_{y} (s) = [\begin{matrix} 1 & 0 \\ s & 1 \end{matrix}] \end{matrix}

For example, a vector $v$ multiplied by $(1)$ results in:

v^{'} = H_{x} (s) v = [\begin{matrix} 1 & s \\ 0 & 1 \end{matrix}] [\begin{matrix} v_{x} \\ v_{y} \end{matrix}] = [\begin{matrix} v_{x} + s v_{y} \\ v_{y} \end{matrix}]

3D Shearing

The notation $H_{xy}$ indicates that the $x$ and $y$ coordinates are shifted by the other coordinate, $z$ , i.e.:

\begin{aligned} x^{'} & = x + s z \\ y^{'} & = y + t z \\ z^{'} & = z \end{aligned}

The shearing matrices in 3D are:

\begin{matrix} (3) & H_{xy} (s, t) = [\begin{matrix} 1 & 0 & s \\ 0 & 1 & t \\ 0 & 0 & 1 \end{matrix}] \end{matrix}

\begin{matrix} (4) & H_{xz} (s, t) = [\begin{matrix} 1 & s & 0 \\ 0 & 1 & 0 \\ 0 & t & 1 \end{matrix}] \end{matrix}

\begin{matrix} (5) & H_{yz} (s, t) = [\begin{matrix} 1 & 0 & 0 \\ s & 1 & 0 \\ t & 0 & 1 \end{matrix}] \end{matrix}

For example, a vector $v$ multiplied by $(3)$ results in:

v^{'} = H_{xy} (s, t) v = [\begin{matrix} 1 & 0 & s \\ 0 & 1 & t \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} v_{x} \\ v_{y} \\ v_{z} \end{matrix}] = [\begin{matrix} v_{x} + s v_{z} \\ v_{y} + t v_{z} \\ v_{z} \end{matrix}]

2D Shearing

3D Shearing

References

Translating Objects with a Transformation Matrix

Scaling Objects with a Transformation Matrix

Transformation Matrix