Dilation Around a Point

The intuition behind dilation (homothety) is changing the size of a figure without changing the shape of the figure, e.g. a square with side length 1 changing to a square with side length 5. A homothety is defined as a transformation of space which dilates distances with respect to a fixed point.

Dilation/Homothety

So what is homothety (huh-MA-tha-tee)? Homothety is the mapping of a figure to a similiar figure of a bigger or smaller size.

In other words, given a point $H$ and a nonzero $k$, a homothety with center $H$ and scaling factor $k$ maps point $A$ to a point $A^\prime$ such that points $H$, $A$, and $A^\prime$ are collinear, and ${HA}^\prime=k\cdot HA$, where $HA$ and $HA^\prime$ are signed segments. We denote the homothety as $\mathcal{H}(H, k)$.

Credits to Rajiv Movva for diagram

The center of homothety in the diagram above is $H$. Our original figure (when $k=1$) is $\triangle ABC$. The dotted blue triangle is when $k=2$, meaning that it is twice the size of $\triangle ABC$. This can be represented as $\mathcal{H}(H,2)$. The dotted yellow triangle is when $k=1/2$, so it's a contracted version of $\triangle ABC$.

But what happens when $k$ is a negative number? Then the figure is reflected or flipped over the center of homothety, which is $H$ in this case. The dotted red triangle is when $k=-2$.

Pretend you're at the Yosemite National Park. You suddenly see a mule deer that catches your eyes and you take out your camera. You want to take a good picture of the deer, but the deer is to too far away. So what do you do? You zoom the camera in! We can think of homothety this way; as the zoom feature on a camera. When $k>1$, we zoom in and the figure is enlarged. When $0<k<1$, we zoom out and the figure shrinks.

There are several key facts about dilation that are important to remember:

1. If a homothety with center $H$ maps $A$ to $A'$, then $H$, $A$, $A'$ are collinear.

2. Dilation preserves angles $\left(\angle ABC =\angle DEF\right)$. This means that the dilated figure is similar to the original figure.

3. Dilation is not an isometry. When we enlarge and shrink a figure, the distances are changed $\left(AB\ne DE\right)$.

4. Dilation preserves ratios $\left(\dfrac{AB}{BC}=\dfrac{DE}{EF}\right)$.

5. Dilation maps a line to a parallel line (In the diagram we have $AB\parallel DE$).

6. Homothety preserves orientation.

7. Homothety maps circles into circles.

8. Homothety preserves tangency.

9. Every homothety has an inverse transformation. If the center of dilation is $H$ and a figure is scaled by a factor of $k$, then $\mathcal{H}^{-1}$ has center $H$ and factor of $\dfrac{1}{k}$.

10. Homothety preserves orientation.

11. If there are two figures such that they are both equally oriented and similar, then there exists a homothety that maps one to the other.

Dilation is the idea of scaling objects to become larger or smaller, and is very similar to the notion of "zooming in and out" of a diagram. So with a dilation, all lengths are increased or decreased at a certain rate. This results in all lengths of a diagram being multiplied by a certain dilation factor, which is completely determined by the given dilation. In other words, a dilation scales every affected length by the same amount.

Dilations are also allowed to be "negative", though this would result in the scaled diagram being "flipped". The result is the same as if you had rotated the diagram by 180 degrees and then scaled. Distances in this case are multiplied by the absolute value of the dilation factor.

A dilation is where an object gets resized to form an image, the resulting shape. The image is similar to the object, thus has a scale factor/ratio of distances.