Translation

Definition

• Translation moves every point of an object a constant distance in a specified direction.
• It is represented by adding the translation vector to the position vectors of the object’s points.

Example of a Translation Matrix:

$$\text{Translate:}\left[\begin{array}{cccc}1& 0& 0& \Delta x\\ 0& 1& 0& \Delta y\\ 0& 0& 1& \Delta z\\ 0& 0& 0& 1\end{array}\right]$$

Scenario 1: Moving an Object Upwards in the Y-Axis

Imagine we have a game object initially located at the coordinates (2, 3, 4) in a 3D game world. We want to move this object 5 units upwards along the Y-axis.

• Translation Vector: $\Delta x=0$, $\Delta y=5$, $\Delta z=0$
• Initial Position Vector: $P_{initial}=[2,3,4]$

Translation Operation: To calculate the new position, we apply the translation matrix to the object’s initial position vector:

$$\text{Translate:}\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 5\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

New Position: $P_{new}=P_{initial}+\Delta P=[2,8,4]$

Scenario 2: Moving an Object Along All Axes

For a more complex example, imagine we want to move the same game object from its initial position to a new position by translating it 3 units along the X-axis, 5 units along the Y-axis, and 2 units along the Z-axis.

• Translation Vector: $\Delta x=3$, $\Delta y=5$, $\Delta z=2$
• Initial Position Vector: $P_{initial}=[2,3,4]$

Translation Operation: Applying the translation matrix to the object’s initial position vector, with our specific $\Delta x$, $\Delta y$, and $\Delta z$ values:

$$\text{Translate:}\left[\begin{array}{cccc}1& 0& 0& 3\\ 0& 1& 0& 5\\ 0& 0& 1& 2\\ 0& 0& 0& 1\end{array}\right]$$

New Position: $P_{new}=P_{initial}+\Delta P=[5,8,6]$