Dilations

Dilation Meaning

Dilation is a transformation technique that is used to make figures either larger or smaller without changing or distorting the shape.

The change in size is done with a quantity called the scale factor. This change in size can be a decrease or increase depending on the scale factor used in the question and is done around a given center point. The images below show enlargement and then a reduction of a shape around the origin.

Fig. 1. Example showing enlargement.

Fig. 2. Example showing a reduction.

Properties of Dilation

Dilation is a non-isometric transformation and as with all transformations uses the notation of pre-image (the original shape) and image (the shape after transformation).

Being non-isometric means that this transformation changes size, however, it will keep the same shape.

Key features of dilated images with regards to their pre-images are,

• All the angles of the dilated image with regard to the pre-image remain the same.
• Lines that are parallel and perpendicular remain so even in the dilated image.
• The midpoint of the side of a dilated image is the same as that in the pre-image.

Dilation Scale Factor

The way we apply dilation is by taking a pre-image and changing the coordinates of its vertices by a scale factor \((r)\) given in the question.

We change the coordinates from a given centre point. We can tell how the image is going to change with regard to the preimage by examining the scale factor. This is governed by,

• The image is enlarged if the absolute scale factor is more than 1.
• The image shrinks if the absolute scale factor is between 0 and 1.
• The image stays the same if the scale factor is 1.

Dilation Formula

We distinguish two cases depending on the position of the center point.

Case 1. The center point is the origin.

The formula to calculate a dilation is direct if our center point is the origin. All we will do is take the coordinates of the pre-image and multiply them by the scale factor.

As seen in the example above, for a scale factor of \(2\) we multiply each coordinate by \(2\) to get the coordinates of each of the image vertices.

Case 2. The center point is not the origin.

But what if our center point isn't the origin? The way we would go about this would be by using a vector to each vertex from the center point and applying the scale factor. Let's consider this in the image below.

Fig. 4. Graphic to demonstrate vector approach.

As you can see in the image above, we aren't given coordinates but vectors from the center point to each vertex. If your center point isn't around the origin this method is the way to solve your dilation problem.

In the image above, we have the centre point at the origin for ease of calculation of the position vector between the centre point and a vertex. But let's consider the image below to see how we could calculate this vector from the centre point.

Fig. 5. Graphic showing how to find position vectors.

In this image, we have one vertex and the centre point for the simplification of the process. When applying this method to a shape, we would repeat the process between the centre point and every vertex.

To find our vector between the centre point and the vertex, we start at our centre point and count how many units the vertex is away from the centre point horizontally to find our \(x\) value. If the vertex is to the right of the centre point we take this as positive, if to the left then negative. Then we do the same but vertically for the \(y\), taking upwards as positive and downwards as negative. In this case, the vertex is 4 units right and 4 units up from the centre point giving the position vector of \(\begin{bmatrix}4\\4\end{bmatrix}\).

We would multiply then each vector by the scale factor to get a vector to each vertex of the image.

If an example of a scale factor was \(1.25\), we would multiply each vector component by \(1.25\) and then from the center point plot this new vector. Once we do this for each vector to the pre-image vertices we would have vectors leading to each vertex of the image.

In terms of notation for a general form let,

• \(C\) = Centre point
• \(A\) = Vertex of pre-image
• \(\vec{CA}\) = Vector from centre point to preimage vertex
• \(r\) = Scale factor
• \(A'\) = Vertex of image
• \(\vec{CA'}\) = vector from centre point to image vertex

The mathematical equation for dilation will therefore be,\[\vec{CA'}=r\cdot \vec{CA}.\]

Dilation Examples

So now we understand how dilation works so let's have a look at a few examples to put the theory into practice.

Origin centre

We'll first examine an example where the centre point is located at the origin.

Positive scale factor

Let's now have a look at a simple example with a positive scale factor and a centre not at the origin.

Negative scale factor

Now we've seen how to apply a positive scale factor but what about if you had a negative scale factor? Let's see what this would look like.

Working back to scale factor

Ok, we know how to perform dilations using scale factors now but what if we aren't given a scale factor but the coordinates of the centre point, image and pre-image? What would this look like?