Polar Curves
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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)
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.

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