I've been curious about Archimedean Spirals and their relations to Sacks Spirals and prime numbers.
I would like to draw some visualizations of the points with a given distance from the center, across the spiral path. Is there a formula for that?
Clarification:
I'm looking for a function that, given a distance along the path, fives the coordinates of a point along that path.
Consider the example image above. $f(0)=(0,0)$, because a distance of 0 along the path is the origin of axis. In this (rough) sketch, $f(1)≃(1, -3)$, $f(2)≃(0.8, 1.1)$, $f(3)≃(-1.3, 1.1)$ and so on.
Thanks a lot for your help!
The arc length for $r=a\theta^{1/n}$ is given by $$ s(\theta)=a\theta^{1/n} {_2F_1}\left(-\frac{1}{2},\frac{1}{2n}; 1+\frac{1}{2n}; -n^2\theta^2\right), $$ where $_2F_1(a,b;c;x)$ is a hypergeometric function. See here.
For $n=1$ (your case) this boils down to $$ \frac{a}{2}\left[\theta\cdot\sqrt{1+\theta^2}+\ln \left(\theta+\sqrt{1+\theta^2} \right)\right]. $$
You get there by using $\vec r= (a\cdot \theta\cdot\cos\,\theta,a\cdot \theta\cdot\sin\,\theta)$ and the formula for the arc length $$ \begin{eqnarray} s(\theta)&=&\int_{0}^{\theta} \sqrt { [r_x'(\varphi)]^2 + [r_y'(\varphi)]^2 }\, d\varphi\\ &=&\int_{0}^{\theta} \sqrt {1+x^2} \, dx .\\ \end{eqnarray} $$ You'll find some ways to solve similar things like the last integral here.