TRANSLATION TRANSFORMATION MATRIX

Translation transformation matrix is the matrix which can be used to make translation transformation of a figure.

Translation - Definition

A translation slides an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.  It is a direct isometry.

In simple words, translation means, it just moves the given figure without rotating, re sizing or anything else.   

Translation Rule Using Matrix

Let A (-2, 1), B (2, 4) and (4, 2) be the three vertices of a triangle. First we have to write the given vertices in matrix form as given below.

If the given figure is translated "h" units on the x-axis and "k" units on the y-axis, then the translation transformation matrix will be as given below.

And we can get the vertices of the translated image using the matrix shown below.

Vertices of the translated image  are

(-2+h, 1+k), (2+h, 4+k), (4+h, 2+k)

Translation for (h ,  k) = (-1 , 2)

Once students understand the above mentioned rule which they have to apply for translation transformation, they can easily make translation-transformation of a figure.

For example, if we are going to make translation transformation of the point (5, 3)  for (h, k) = (1, 2), after transformation, the point would be (6, 5).

Here, the rule we have applied is

(x, y) ----> (x + h, y + k) 

So we get

(5, 3) ----> (6, 5)

Let us consider the following example to have better understanding of translation transformation matrix.

Question :

Let A(-2, 1), B(2, 4) and C(4, 2) be the three vertices of a triangle. If this triangle is translated for (h, k) = (-1, 2) what will be the new vertices A', B' and C' ?

Solution :

Step 1 :

First we have to write the vertices of the given triangle ABC  in matrix form as given below.

Step 2 :

Since the triangle ABC is translated for (h, k) = (-1, 2), we have to find the vertices of the translated image using matrices as given below. 

Step 4 :

Now we  can get vertices of the translated image A'B'C' from the resultant matrix. 

Vertices of the translated image are

A'(-3, 3), B'(1, 6) and C'(3, 4)

How to sketch the translated figure?

1. First we have to plot the vertices of the pre-image.

2. In the above problem, the vertices of the pre-image are

A(-2, 1) , B(2, 4) and C(4, 2)

3. When we plot these points on a graph paper, we will get the figure of the pre-image (original figure).

4. When we translate the given figure for (h, k)  =  (2, 3), we have to use the rule using matrices.

5. When we apply the method explained above, we will get the following vertices of the image (translated figure).

6. In the above problem, vertices of the image are

A'(-3, 3), B'(1, 6) and C'(3, 4)

7. When plot these points on the graph paper, we will get the figure of the image (translated figure).

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