Translating Objects with a Transformation Matrix

We build different types of transformation matrices to translate objects along cardinal axes and arbitrary axes in 2D and 3D with matrix multiplication!

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

This article is part 4 in the series about transformation matrices:

2D Translation

A translation is an affine transformation, which is a linear transformation followed by some displacement.

v^{'} = Mv + b

Even though we can’t express 2D translation using a 2x2 matrix, we can express such a transformation as a shearing transformation in 3D projective geometry . To do so, we have to imagine that the 2D Euclidean world exists as the plane $w = 1$ in a 3D space. Under this geometry, any point has the form $[\begin{matrix} x & y & 1 \end{matrix}]$ .

In Euclidean geometry, a vector expressed as a linear combination of the standard basis has the form:

v = v_{x} \hat{i} + v_{y} \hat{j} = {[\begin{matrix} v_{x} & v_{y} \end{matrix}]}^{T}

In projective geometry, a vector that exists in the plane $w = 1$ has the form:

v = v_{x} \hat{i} + v_{y} \hat{j} + 1 \hat{w} = {[\begin{matrix} v_{x} & v_{y} & 1 \end{matrix}]}^{T}

This basis can be represented using the following transformation matrix:

M = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} \overset{|}{\underset{|}{i}} & \overset{|}{\underset{|}{j}} & \overset{|}{\underset{|}{w}} \end{matrix}]

The translation transform then can be seen in projective geometry as a simple shearing of the space by the coordinate $w$ , using the shearing transform $H_{xy} (Δ x, Δ y)$ to transform a point $v$ :

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

Now that we’re using perspective geometry to represent entities, let’s imagine a point $p = [\begin{matrix} x & y & 0 \end{matrix}]$ (a point that lies in the plane $w = 0$ ). Whenever this point is transformed by a transformation matrix, we can notice that the translation components of the matrix are canceled because of $w = 0$ . We can take advantage of this fact and represent vectors with this notation.

Let $v_{\infty}$ be a point located in the plane $w = 0$ . Applying the shearing operation $H_{xy} (s, t)$ results in:

v_{\infty}^{'} = H_{x y} (Δ x, Δ y) v_{\infty} = [\begin{matrix} 1 & 0 & Δ x \\ 0 & 1 & Δ y \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} v_{x} \\ v_{y} \\ 0 \end{matrix}] = [\begin{matrix} v_{x} \\ v_{y} \\ 0 \end{matrix}]

It’s important to note that this matrix multiplication is still a linear transformation and that this trick of translating 2D points is actually a shearing of the 3D projective plane.

3D Translation

Similarly to 2D, a 3D translation can be represented as a shearing of the 4D projective hyperplane, which has the form:

T = H_{xyz} (Δ x, Δ y, Δ z) = [\begin{matrix} 1 & 0 & 0 & Δ x \\ 0 & 1 & 0 & Δ y \\ 0 & 0 & 1 & Δ z \\ 0 & 0 & 0 & 1 \end{matrix}]

When a 4D vector existing on the hyperplane $w = 1$ is transformed with this matrix, the result is:

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

The general 3D translation matrix is then denoted as:

\begin{matrix} (1) & T = [\begin{matrix} 1 & 0 & 0 & T_{x} \\ 0 & 1 & 0 & T_{y} \\ 0 & 0 & 1 & T_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] = [\begin{matrix} I_{3 \times 3} & T_{3 \times 1} \\ 0_{1 \times 3} & 1 \end{matrix}] \end{matrix}

2D Translation

3D Translation

References

Shearing Objects with a Transformation Matrix

Scaling Objects with a Transformation Matrix

Transformation Matrix