'Loops' / "Arcs" refers to PI RAD / 180 deg
Sorry for the long title. I was playing around with hyperbolic Spirals today, and noted, as you all know, that the difference between the size of their 'arcs' / 'loops' decreases towards the center, i.e., the 'arc' / 'loops' get more similar as the graph gets closer to 0 (see fig). This happens 'hyperbolic-ally.' So, I naturally wondered if it's possible to do the same thing via a constant, for instance,
$$0.4^{0}*(Difference)$$ $$0.4^{1}*(Difference)$$ $$0.4^{2}*(Difference)$$ $$0.4^{3}*(Difference)$$ $$0.4^{4}*(Difference)$$
And so on.
This seems possible, but I can't quite get it and would be thankful for help!
If the difference in radii between consecutive arcs of the spiral were to increase by a constant amount with each 180 degree turn, then the radius would be increasing at a rate of some constant k>0 per $\pi$ radians. This relationship can be written in polar coordinates as r=$\frac{k\theta}{\pi}$. Using the the transformation from polar to Cartesian coordinates: $(r,\theta)\rightarrow(rcos(\theta),rsin(\theta))$, we can parameterize the spiral as $s(\theta)=(\frac{k\theta}{\pi}cos(\theta),\frac{k\theta}{\pi}sin(\theta))$.