How to place objects equidistantly on an Archimedean spiral?

To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects.

This is what I have so far: The length of a spiral is determined by $$ l = \frac{a}{2}\left[\varphi\cdot\sqrt{1+\varphi^2}+\ln \left(\varphi+\sqrt{1+\varphi^2} \right)\right] $$ I presume that solving this equation for $\varphi$ will give me what I need. But trying that with WolframAlpha leads to a timeout.

Is solving this equation for $\varphi$ really the right thing to do? If yes, how can I solve it?

Solution 1:

In order to place points equidistantly according to the arc-length, you should place points at $\phi_k$ determined by $$ x_k = k \frac{\Delta \ell}{2 a} = \phi_k \cdot \sqrt{1+\phi_k^2} + \operatorname{arcsinh}(\phi_k) $$ This equation admits no solution in simple functions, but can be easily solved numerically.

Also for large $x_k$, $\phi_k \sim \sqrt{x_k}$. More precisely: $$ \phi_k \sim \sqrt{ \frac{1}{2} W\left( \frac{1}{2} \mathrm{e}^{2x_k -1}\right)} $$ where $W(x)$ is Lambert W function. This gives rather good placement (red rod represent approximate locations, and centers of the blue circles represent exact solutions):