# Polar Curves

Polar curves are defined by $r = f(\theta)$, or rather, all points $P$ that have at least one polar representation of $(r, \theta)$ whose coordinates will satisfy the equation.

We will first look at some elementary examples, such as $r = 1$. This can clearly be written as $r = 1 + 0\theta$ to have both $r$ and $\theta$ in the equation. We thus get a curve of all points $P$ that are of length $1$ away from the origin, and revolve around the circle formed by rotating the ray $r$ of length $1$ as $\theta$ varies. Hence thus get the following circle:

Let's look at the example of $\theta = 1$. Once again this can be rewritten as $\theta = 1 + 0r$. So we will plot all points P that have an angle of $\theta$ being $1$ radian, with $r$ (the position from the origin) varying. We hence obtain the following curve:

This curve is clearly a line, and it should make sense as to why. As each point $P$ on the line has the same angle of $1$ radian, while their distance away from the origin, that is their $r$ (the length of the ray) increases.

Now let's look at some examples of polar curves that are more difficult

## Example 1

**Sketch the polar curve $r = \sin \theta$ on the interval $0 ≤ \theta ≤ 2\pi$**

To begin sketching the curve, let's first look at the curve $r = \sin \theta$ on a $r-\theta$-coordinate system (which is analogous to the $xy$-coordinate system):

Let's find some points to plot by setting $\theta$ equal to various angles:

Θ | r | (r, Θ ) |
---|---|---|

0 | 0 | (0, 0) |

π/4 | √2/2 | (√2/2, π/4) |

π/2 | 1 | (1, π/2) |

3π/4 | √2/2 | (√2/2, 3π/4) |

π | 0 | (0, π) |

5π/4 | -√2/2 | (-√2/2, 5π/4) |

3π/2 | -1 | (-1, 3π/2) |

7π/4 | -√2/2 | (-√2/2, 7π/4) |

2π | 0 | (0, 2π) |

So we are going to start at the origin $(0,0)$ and have the distance the the point it from the origin increase from the interval $(0, \frac{\pi}{2})$ as the ray length is increasing as theta increases.

Then on the interval $(\frac{\pi}{2}, \pi)$, the point tracing the curve is going to return to the origin at $(0, \pi)$.

On the interval $(\pi, \frac{3\pi}{2})$, $r$ becomes negative, so the point the curve subtracts $\pi$ from each angle to make $r$ positive, and hence, the first part of the curve from $(0, \frac{\pi}{2})$ is retraced. The same happens on the interval $(\frac{3\pi}{2}, 2\pi)$. Hence the graph of the curve in polar coordinates is as follows:

Hence a circle is drawn. If were were to not restrict the domain to $[0, 2\pi]$, then we would trace the circle out infinite times.