Polar curve r proportional to theta
Categories: polar coordinates
Here is a polar function where $r$ is proportional to $\theta$.
$$r = \theta$$
Describes a curve where $r$ is constant for any angle $\theta$, in this case 3. The curve includes 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 $2\pi$:
Scaling the curve
In general the function:
$$r = a \theta$$
where $a$ is a non-negative constant, describes a spiral curve of that is similar to the curve above (ie it is the same shape, but scaled by $a$). Here are the curves created for $a = 1$ (red), $a = 1/2$ (green), and $a = 1/4$ (blue):
The curve will cross the x-axis at 0, $-a\pi$ and $2 a\pi$. It will cross the y-axis at $a \frac{\pi}{2}$ and $-3 a \frac{\pi}{2}$.
See also
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