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 $r = -4\cos \theta$ |

$(-0.5, 0)$ | $(-1, 0)$ | $(-2,0)$ |