2.1. Translation

Moves a point by a specified distance in $x$ and $y$.

Translation is a basic geometric transformation that shifts a point or shape from one location to another in a 2D space. If we have a point (x, y), translation involves adding specific values, known as the translation distance, to both the x and y coordinates. For example, to translate a point (2, 3) by a distance of 4 units to the right and 2 units up, we will perform the calculation:

• New x-coordinate: 2 + 4 = 6
• New y-coordinate: 3 + 2 = 5 Thus, the point (2, 3) translates to (6, 5).

Imagine you have a toy car on a table. If you want to move the car 4 inches to the right and 2 inches upwards, you're performing a translation. Just as you would physically move the car on the table, in coordinate geometry, we are adjusting the point's position by adding distances to its current coordinates.

Transformation Matrix: $T \cdot$

In the context of translation, we often represent these shifts using a transformation matrix. A transformation matrix is a mathematical tool that allows us to apply geometric transformations like translation, scaling, rotation, and others. For a 2D translation, we typically use a 3x3 matrix. The general form of the translation matrix looks something like this:

$T = \begin{bmatrix} 1 & 0 & tx \\ 0 & 1 & ty \\ 0 & 0 & 1 \end{bmatrix}$

• Here, tx is the translation distance along the x-axis, and ty is the distance along the y-axis. When we multiply this matrix with a point expressed in homogeneous coordinates (x, y, 1), we get the translated point.

Think of the transformation matrix as a set of instructions for a delivery robot. Just like you would give the robot directions to move to a specific location (like ‘go 4 meters right and 2 meters up’), the transformation matrix provides the specific numerical instructions on how to alter the point's location in a coordinate system.

Applied as: $T \cdot$

To apply the translation, we take a point expressed in homogeneous coordinates, which includes an additional dimension (w = 1), and multiply it by the translation matrix. This process effectively recalculates the new location of the point in 2D space. For example, if the point we want to translate is (1, 2), we express it in homogeneous coordinates as (1, 2, 1). When we multiply this by our translation matrix T, we obtain the new position:

$\begin{bmatrix} 1 & 0 & tx \\ 0 & 1 & ty \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$ This matrix multiplication gives us the translated coordinates directly.

Imagine you are a map maker and the translation matrix is a specific route you mark out on your map. By following your route (or multiplying with the matrix), you navigate to a new location on the map effortlessly, just as the original coordinates become new coordinates through the application of the translation matrix. It’s like using GPS coordinates to get from one place to another.

--