I'm attempting to generate a Archimedes spiral (defined as $r = a\theta$) from a given width $w$ and spacing $a$ between 'arms'. I have plotted Cartesian coordinates generated from my workings, but the values don't appear correct (~4 orders of magnitude out).
I'm absolutely certain I'm making a ridiculous mistake that is blindly obvious to everyone else. (EDIT: I'm an idiot. turns out there was a rogue divide by 1000 in there. The spacing and max $\theta$ still look off though.)
I have defined the maximum width $w$ as the sum of $r$ and it's opposite, one turn behind on the spiral ($w = a\theta+a(\theta-\pi)$), rearranged for $\theta$ ($\theta = w/2a + \pi/2)$), and converted to Cartesian equations ($x = a\theta cos\theta, y = a\theta sin\theta$).
To generate points, I am subdividing the max $\theta$ by 1000 and calculating an $(x,y)$ for each value. For a width of 1000 and spacing of 100 I'm getting a spiral with about 8 turns, which makes the spacing about 625.
I have read up on polar coordinates and Archimedean spirals for this but I am stuck at this point, I'm unsure on how to check my working. I have found a lot of info on arc length of a spiral, but nothing on the width. Any help appreciated.
For posterity: An Archimedean spiral $r = a\theta$ may be represented parametrically by $$ x = a\theta \cos\theta,\qquad y = a\theta \sin\theta. \tag{1} $$ The radial separation between successive windings is $s = 2\pi a$.
If (1) is plotted over $n$ turns starting at $\theta = 0$, i.e., for $0 \leq \theta \leq 2n\pi$, the width $w$ is \begin{align*} w &= ns + (n - \tfrac{1}{2})s \\ &= (2n - \tfrac{1}{2})s \\ &= (4n - 1)\pi a. \end{align*}