Have you ever wondered how your phone allows you to zoom in on pictures to blow the image up? What would this process be called and how would it work?
Well, this is an application of dilation- you are enlarging an image around a center point (where you started zooming from) by a factor driven by how much you move your fingers.
Read on to find out more about how this transformation works!
Dilation Meaning
Dilation is a transformation that resizes a pre-image, it is therefore non-isometric.
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 scale factor is the ratio of the size of the image to the size of the pre-image. It is calculated as, \[\mbox{scale factor} = \frac{\mbox{dimensions of image}}{\mbox{dimensions of pre-image}}.\]
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.
The scale factor can't be equal to 0.
If we had a scale factor of \(2\), the vertices of the image would each be double the distance away from the center point than the preimage and would therefore be larger.
Inversely, a scale factor of \(0.5\) would mean each vertex would be closer by half to the center point than the preimages vertices.
A scale factor of \(2\) is shown below on the left, and a scale factor of \(0.5\) on the right. The center point for both images is the origin and is labeled G.
Fig. 3. Graphic showing how the scale factor affects the image around a center point.
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.
Consider a square with vertices located at \((4,4)\), \((-4,4)\), \((-4,-4)\) and \((4,-4)\). The centre point is at the origin and the scale factor is \(r=1.5\). Sketch the image on a graph.
Solution
First, we sketch what we know from the question as seen below.
Fig. 6. Pre-image set up.
Since we are based around the origin, all we have to do is multiply the coordinates by the scale factor to receive the new coordinates. We only have \(4\) or \(-4\) as our coordinates so these will each become \(6\) or \(-6\) respectively as \(4\cdot 1.5=6\) and \(-4\cdot 1.5=-6\). This would result in the image seen below.
Fig. 7. Final image sketch.
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.
Consider a triangle with vertices located at \(X=(0,3)\quad Y=(2,4)\quad Z=(5,2)\).
The centre point is defined as \(C=(-1,-1)\) and the scale factor is \(r=0.75\). Sketch the pre-image and image on a graph.
Solution
Our first step will be to sketch the pre-image and the center point and define our vectors to each vertex.
Examining the coordinates we can see that to move from the centre point to \(X\), we must move \(1\) right and \(4\) up. This is as \(-1\) to \(0\) increases by one, and \(-1\) to \(3\) increases by four. To move to \(Y\) we move \(3\) right and \(5\) up, and to \(Z\) we move \(6\) right and \(3\) up.
Fig. 8. Sketch of pre-image, centre point and vectors to each vertex.
So now we have our first sketch, all we need to do is apply the formula seen earlier to each vertex.\[\begin{align}\vec{CX'}&=r\cdot \vec{u}\\&=0.75\cdot \begin{bmatrix}1\\4\end{bmatrix}\\&=\begin{bmatrix}0.75\\3\end{bmatrix}\end{align}\]
\[\begin{align}\vec{CY'}&=r\cdot \vec{v}\\&=0.75\cdot \begin{bmatrix}3\\5\end{bmatrix}\\&=\begin{bmatrix}2.25\\3.75\end{bmatrix}\end{align}\]
\[\begin{align}\vec{CZ'}&=r\cdot \vec{w}\\&=0.75\cdot \begin{bmatrix}6\\3\end{bmatrix}\\&=\begin{bmatrix}4.5\\2.25\end{bmatrix}\end{align}\]
Having our new position vectors scaled by our scale factor, we can now sketch our image.
From the centre point of \((-1,-1)\) we will move \(\begin{bmatrix}0.75\\3\end{bmatrix}\) to give the coordinates of \(X'\) as \((-0.25,2)\) from the calculation:\[x=-1+0.75=-0.25\]\[y=-1+3=2\]
For \(Y'\):\[x=-1+2.25=1.25\]\[y=-1+3.75=2.75\]\[Y'=(1.25,2.75)\]
For \(Z'\):\[x=-1+4.5=3.5\]\[y=-1+2.25=1.25\]\[Z'=(3.5,1.25)\]
We then plot our new vertices, and we obtain the below image. We notice that the image is sized down as the scale factor is less than 1.
Fig. 9. Sketch of image and pre-image.
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.
Consider a triangle with vertices located at \(X=(0,3)\quad Y=(2,4)\quad Z=(5,2)\). The centre point is defined as \(C=(-1,-1)\) and the scale factor is \(r=-2\). Sketch the pre-image and image on a graph.
Solution
Our first sketch of setting up the question is the same as the last example. Therefore see the graph below,
Fig. 10. Initial sketch set up.
Now we will apply the same mathematic formulas as last time to get our new vectors but this time \(r=-2\):
\[\begin{align}\vec{CX'}&=r\cdot \vec{u}\\&=-2\cdot \begin{bmatrix}1\\4\end{bmatrix}\\&=\begin{bmatrix}-2\\-8\end{bmatrix}\end{align}\]
\[\begin{align}\vec{CY'}&=r\cdot \vec{v}\\&=-2\cdot \begin{bmatrix}3\\5\end{bmatrix}\\&=\begin{bmatrix}-6\\-10\end{bmatrix}\end{align}\]
\[\begin{align}\vec{CZ'}&=r\cdot \vec{w}\\&=-2\cdot \begin{bmatrix}6\\3\end{bmatrix}\\&=\begin{bmatrix}-12\\-6\end{bmatrix}\end{align}\]
Having our new position vectors scaled by our scale factor, we can now sketch our image.
From the centre point of \((-1,-1)\) we will move \(\begin{bmatrix}-2\\-8\end{bmatrix}\) to give the coordinates of \(X'\) as \((-3,-9)\) from the calculation:
\[x=-1-2=-3\]
\[y=-1-8=-9\]
For \(Y'\):
\[x=-1-6=-7\]
\[y=-1-10=-11\]
\[Y'=(-7,-11)\]
For \(Z'\):
\[x=-1-12=-13\]
\[y=-1-6=-7\]
\[Z'=(-13,-7)\]
Fig. 11. Sketch with negative scale factor.
As you can see in the image above, when we have a negative scale factor we apply the same principle as a positive scale factor. The only difference is the image ends up on the other side of the centre point.
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?
You have a pre-image with the coordinates \(X=(1,5)\quad Y=(2,3)\quad Z=(4,-1)\) and an image with the coordinates \(X'=(3,15)\quad Y'=(6,9)\quad Z'=(12,-3)\). What is the scale factor of the dilation?SolutionWe know that the scale factor can be defined as seen below:\[\mbox{scale factor} = \frac{\mbox{dimensions of image}}{\mbox{dimensions of pre-image}}.\]Therefore, if we find the ratio between an image dimension and a pre-image dimension we will have the scale factor. Let's do this with the \(x\) component of the \(X\) coordinates.\[\begin{align}\mbox{scale factor} &= \frac{\mbox{dimensions of image}}{\mbox{dimensions of pre-image}}\\&=\frac{3}{1}\\&=3\end{align}\]This gives the scale factor of the transformation. Let's check this with the \(x\) component of the \(Z\) variable.\[\begin{align}\mbox{scale factor} &= \frac{\mbox{dimensions of image}}{\mbox{dimensions of pre-image}}\\&=\frac{12}{4}\\&=3\end{align}\]This check shows our original calculation was correct and the scale factor of the transformation is given as \(r=3\).
Dilations - Key takeaways
Dilation is a non-isometric transformation and is the resizing of an image, driven by a scale factor and centre point.
The scale factor is defined as:\[\mbox{scale factor} = \frac{\mbox{dimensions of image}}{\mbox{dimensions of pre-image}}.\]
If the absolute value of the scale factor is greater than one, the image is enlarged. If the absolute of the scale factor is between 0 and 1 then the image is shrunk.
The vector from the centre point to an image vertex is given as:\[\vec{CA'}=r\cdot \vec{CA},\]where:
- \(C\) = 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
If the scale factor is negative, the image is located on the other side of the centre point and resized by the absolute value of the scale factor.
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