Dilatations happen when you make an image larger or smaller. Its shape and orientation remain the same, but its size and position change. Here are some simple illustrations:
In this example, a rectangle was enlarged to twice its size. Its sides are now twice as long as they were before. This is a dilatation of factor 2.

Here we have a dilation of factor 1/3. Notice that the original shape has been shrunk ... it's sides are now only one third as long.

How To Make A Dilatation:
Let's begin with a simple house shape. We plan to enlarge it by a factor of 2. We've drawn the original shape, and clearly marked the corners. We've also shown the centre of dilatation. This blue dot represents the point from which we will expand the image. Every dilatation requires a centre of dilatation. It will be the centre point from which we construct the lines necessary to find the image.
The next step will to draw lines from the centre of dilatation through each of the corners of the figure, as shown here. Since this is to be an enlargement by factor 2, we must also identify the point on the image that is twice as far from the centre as the original corner. We've shown that point here as a green dot.
We'll need to do this for all the corners of the original shape. Draw the lines carefully with a ruler, and measure distances as exactly as you can.
O.K., let's start.
We started over at the top, and drew lines from the centre of dilatation through all the corners of the shape. We marked the first image point with a green dot. Notice that it is twice as far from the centre of dilatation as the original point. In other words, distance d2 is equal to distance d1.
Now you have to locate all the other corresponding image points.
Here is the diagram with all of the points plotted with green dots (each one is twice as far from the centre of dilatation as its corresponding original point). We've also connected the image points to show the enlarged image.
If you do this correctly, the image should have the same shape and orientation as the original figure.
Here is the final diagram. You can check that you did everything correctly by measuring the lengths of the sides of the image. Its sides should all be twice as long as the corresponding sides on the original figure.
We've shown that here by measuring the left side of the roof. The image length L2 is exactly twice as long as the original length L1.

Now that you know how dilatations work, let's look at what happens when you do one on a set of xy axes where, the centre of dilatation is the point (0,0). Go on to page 2.

Mr. Willis
