Here is an example of a polar function of constant $r$.
$$r = 3$$
This function describes a curve where $r$ is constant for any angle $\theta$, in this case 3. The curve represents the locus all the points that are a distance of exactly 3 from the origin. This curve, of course, is a circle of radius 3:
This animation shows how the curve is plotted as $\theta$ moves from 0 to $2\pi$:
In general the function:
$$r = n$$
where $n$ is a non-negative constant, describes a circle of radius $n$. Here are the circles created for $n = 3$ (red), $n = 2$ (green), and $n = 1$ (blue):
We can convert the polar equation into a Cartesian equation using the identity:
$$r^2 = x^2 + y^2$$
Substituting for $r$ in the equation above gives:
$$x^2 + y^2 = n^2$$
which is the Cartesian equation for a circle of radius $n$, centred on the origin.
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