# Limacon Curves

## Key Questions

• Limaçon curves look like circles. They have various types depending on the values in their equations.

In the image below, for example, the types of limaçon curves are: dimpled, cardioid, and looped (respectively).

It's easiest to watch a video of this.
Here's one I made a while back

#### Explanation:

${r}^{2} = {a}^{2} \cos 2 \theta$.
The graph is the fallen eight, looking like the symbol $\infty$...

#### Explanation:

For the double loop of ${r}^{2} = {a}^{2} \cos 2 \theta , \cos 2 \theta \ge 0$.

So, the range for $\theta \in \left[0 , 2 \pi\right]$ excludes $\left(\frac{\pi}{4} , \frac{3 \pi}{4}\right)$ and

$\left(\frac{5 \pi}{4} , \frac{7 \pi}{4}\right)$, Here, $\cos 2 \theta < 0$...