# Families of Polar Curves

We will now look at the following families of polar curves:

- $r = a \tan \theta$.

- $r = \tan (b \theta)$.

## Family of r = atanΘ

Let's now look at the family of polar curves defined by $r = a \tan \theta$ starting with $a > 0$, and a being an integer

$r = \tan(\theta)$ | $r = 2\tan(\theta)$ | $r = 4\tan(\theta)$ |
---|---|---|

As you can see, as a increases, the width of "branches" on the curve increases similarly to that of a parabola.

Now let's look at the family of curves with $a < 0$.

$r = -\tan(\theta)$ | $r = -2\tan(\theta)$ | $r = -4\tan(\theta)$ |
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These curves are completely identical. You can imagine the transformation of $r = b \tan \theta$ to $r = -b\tan \theta$ as a reflection over the $x$-axis or polar-axis, in which case both of these graphs of symmetric amongst the $x$-axis, hence they look the same.

## Family of r = tan(bΘ)

We're now going to look at the family of curves in the form of $r = \tan(b \theta)$ where $b > 0$.

$r = \tan(\theta)$ | $r = \tan(2\theta)$ | $r = \tan(4\theta)$ |
---|---|---|

Notice that as $b$ increases, the number of "branches" for the polar curves increases. In fact for $b$ being a positive integer, there will always be exactly $2b$ branches for this family of curves.

Now let's look at the case where $b < 0$.

$r = \tan(-\theta)$ | $r = \tan(-2\theta)$ | $r = \tan(-4\theta)$ |
---|---|---|

Once again these curves are identical to the ones above with the same property of the number of branches applying for $b$ as an integer and $b \neq 0$. Once again, they're identical as a reflection amongst the $x$-axis results in the same graph.