Families of Polar Curves

# Families of Polar Curves

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

• $r = a \sin \theta$.
• $r = a \cos \theta$.

## Families of r = asinθ

We have already examined the polar curve $r = \sin \theta$, but now let's examine the curves $r = a \sin \theta$ where $r > 0$, and $a \in \mathbb{R}$

$r = \sin \theta$ $r = 2\sin \theta$ $r = 4\sin \theta$
Radius of $r = \sin \theta$ Radius of $r = 2\sin \theta$ Radius of $r = 4\sin \theta$
$0.5$ $1$ $2$
Center of $r = \sin \theta$ Center of $r = 2\sin \theta$ Center of $r = 4\sin \theta$
$(0, 0.5)$ $(0, 1)$ $(0,2)$

As a increases, the size of the circle also increases. Now let's look at the same family of curves with $a < 0$.

$r = -\sin \theta$ $r = -2\sin \theta$ $r = -4\sin \theta$
Radius of $r = -\sin \theta$ Radius of $r = -2\sin \theta$ Radius of $r = -4\sin \theta$
$0.5$ $1$ $2$
Center of $r = -\sin \theta$ Center of $r = -2\sin \theta$ Center of $r = -4\sin \theta$
$(0, -0.5)$ $(0, -1)$ $(0,2)$

Once again, the circles increase size as $a \to -\infty$, but the circle have essentially been flipped over the $x$-axis or the polar axis.

## Families of r = acosθ

The family of curves $r = a \cos \theta$ for $a > 0$, and $a < 0$ follow similar patterns to the family of curves $r = a \sin \theta$:

$r = \cos \theta$ $r = 2\cos \theta$ $r = 4\cos \theta$
Radius of $r = \cos \theta$ Radius of $r = 2\cos \theta$ Radius of $r = 4\cos \theta$
$0.5$ $1$ $2$
Center of $r = \cos \theta$ Center of $r = 2\cos \theta$ Center of $r = 4\cos \theta$
$(0.5, 0)$ $(1, 0)$ $(2,0)$
$r = -\cos \theta$ $r = -2\cos \theta$ $r = -4\cos \theta$
Radius of $r = -\cos \theta$ Radius of $r = -2\cos \theta$ Radius of $r = -4\cos \theta$
$0.5$ $1$ $2$
Center of $r = -\cos \theta$ Center of $r = -2\cos \theta$ Center of $r = -4\cos \theta$
$(-0.5, 0)$ $(-1, 0)$ $(-2,0)$