Translation

Translation is a linear transformation that moves a set of points by the vector \(\mathbf{t}\) (Fig. 31).

Fig. 31 The translation of a point \(\mathbf{u}\) by the translation vector \(\mathbf{t}\) to the point \(\mathbf{v}\)

Using vector addition it is easy to see that

\[\mathbf{v} = \mathbf{u} + \mathbf{t}.\]

To represent translation using a single transformation matrix we need a matrix \(T(\mathbf{t})\) that satisfies

\[\begin{split} \begin{align*} T(\mathbf{t}) \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix} = \begin{pmatrix} u_1 + t_1 \\ u_2 + t_2 \\ u_3 + t_3 \end{pmatrix}. \end{align*} \end{split}\]

Unfortunately the matrix \(T(\mathbf{t})\) does not exist which is independent of \(u_1\), \(u_2\) and \(u_3\) so we can use a trick which makes use of homogeneous co-ordinates.

Definition 19 (Homogeneous co-ordinates)

The homogeneous co-ordinates of a point \(\mathbf{u}\) in \(\mathbb{R}^3\) expressed using the Cartesian co-ordinates is the 4-tuple \((w x, w y, w z, w)\) where \(w \in \mathbb{R}\backslash \{0\}\).

Note that a point with homogeneous co-ordinates \((x, y, z, 1)\) has the Cartesian co-ordinates \((x, y, z)\).

If \(\mathbf{u}= (u_x, u_x, u_y, 1)\) expressed using homogeneous co-ordinates then the translation by the translation vector \(\mathbf{t} = (t_x, t_y, t_z)\) is

\[\begin{split} \begin{align*} v_x &= u_x + t_x, \\ v_y &= u_y + t_y, \\ v_z &= u_z + t_z, \end{align*} \end{split}\]

which can be written as the matrix equation

\[\begin{split} \begin{align*} \begin{pmatrix} v_x \\ v_y \\ v_z \\ 1 \end{pmatrix} &= \begin{pmatrix} 1 & 0 & 0 & t_x \\ 0 & 1 & 0 & t_y \\ 0 & 0 & 1 & t_z \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} u_x \\ u_y \\ u_z \\ 1 \end{pmatrix}. \end{align*} \end{split}\]

The square matrix is the transformation matrix for translating a point of set of points by the translation vector \(\mathbf{t}\).

Definition 20 (Translation matrix)

The transformation matrix for the translation of a vector in \(\mathbb{R}^3\) expressed using homogeneous co-ordinates by the translation vector \(\mathbf{t} = (t_x, t_y, t_z)\) is

(22)\[\begin{split}T(\mathbf{t}) = \begin{pmatrix} 1 & 0 & 0 & t_x \\ 0 & 1 & 0 & t_y \\ 0 & 0 & 1 & t_z \\ 0 & 0 & 0 & 1 \end{pmatrix}\end{split}\]

The transformation matrix for inverse translation is

(23)\[\begin{split}T^{-1}(\mathbf{t}) = T(\mathbf{-t}) = \begin{pmatrix} 1 & 0 & 0 & -t_x \\ 0 & 1 & 0 & -t_y \\ 0 & 0 & 1 & -t_z \\ 0 & 0 & 0 & 1 \end{pmatrix}\end{split}\]

Example 23

A triangle is defined by three points with position vectors \(\mathbf{p}_1 = (1, 0, 1)\), \(\mathbf{p}_2 = (3, 0, 1)\) and \(\mathbf{p}_3 = (2, 0, 3)\). The triangle is translated by the translation vector \(\mathbf{t} = (3, 0, 1)\). Calculate the positions of the triangle vertices of the translated triangle.

Solution

The homogeneous co-ordinate matrix is

\[\begin{split} \begin{align*} P = \begin{pmatrix} 1 & 3 & 2 \\ 0 & 0 & 0 \\ 1 & 1 & 3 \\ 1 & 1 & 1 \end{pmatrix}, \end{align*} \end{split}\]

and the translation matrix is

\[\begin{split} \begin{align*} T(3, 0, 1) &= \begin{pmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix}. \end{align*} \end{split}\]

Applying the transformation

\[\begin{split} \begin{align*} T(3,0,1) \cdot P &= \begin{pmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 3 & 2 \\ 0 & 0 & 0 \\ 1 & 1 & 3 \\ 1 & 1 & 1 \end{pmatrix} \\ &= \begin{pmatrix} 4 & 6 & 5 \\ 0 & 0 & 0 \\ 2 & 2 & 4 \\ 1 & 1 & 1 \end{pmatrix}. \end{align*} \end{split}\]

So the vertex co-ordinates of the translated triangle are \((4,0,2)\), \((6, 0, 2)\) and \((5, 0, 4)\). The original triangle and the translated triangle are plotted in Fig. 32 looking along the \(y\)-axis.

MATLAB code

The following MATLAB code applies the translation from Example 23 and plots the original and translated polygons.

% Define homogeneous co-ordinate matrix
P = [ 1, 3, 2 ;
      0, 0, 0 ;
      1, 1, 3 ;
      1, 1, 1 ];

% Define translation matrix 
T = @(t) [ 1, 0, 0, t(1) ; 
           0, 1, 0, t(2) ; 
           0, 0, 1, t(3) ; 
           0, 0, 0, 1 ];

% Apply translation
t = [3 ; 0 ; 1];
P1 = T(t) * P;

% Plot polygons
figure
patch(P(1,:), P(3,:), 'b', FaceAlpha=0.5)
patch(P1(1,:), P1(3,:), 'r', FaceAlpha=0.5)
axis equal
axis([0, 7, 0, 5])
xlabel("$x$", FontSize=12, Interpreter="latex")
ylabel("$z$", FontSize=12, Interpreter="latex")
box on

Fig. 32 Translating a polygon.