Can anyone explain the behavior of this function?

Question

I inserted the function $r=\sin(\theta/v)$, where $v$ was an adjustable constant. Upon reaching each number, it gave a very interesting graph. The way Desmos works is to prevent overflow, it allows you to specify the highest number $\theta$ can be, in units of $\pi$. When the value of $v$ exceeded the upper limit of $\theta$ divided by $\pi$, the graph devolved into an ever-shrinking spiral. To further visualize, here are some graphs. In all of these, the highest value of $\theta$ is $100\pi$. When the value of $v$ is less than $100$, there is no spiral. But when it is greater than $100$ (like $1000$ shown here), it devolves into a packed spiral. Can anyone explain why this function behaves this way?

Answer 1

The graph $r=\theta$ looks like a spiral.

For $\theta$ small, $\sin \theta \approx \theta$. There is a trade-off between the value of $v$ and how far out you plot it. Eventually, this approximation breaks down and the graph no longer looks like a spiral.

Answer 2

$$ r = \sin\frac{\theta}{v}$$

By differentiation

Radial component of spiral

$$ \Delta r = \frac{\Delta \theta}{v}\cos\frac{\theta}{v}$$

Circumferential component

$$ r \Delta {\theta}= \Delta \theta \sin\frac{\theta}{v}$$

At high $\theta$ spiral increments gets close. At low $\theta$ there is a larger radial component.