# Width of a spiral

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.

• Is it possible you're computing trig functions in "degrees mode", while your width formula (containing $\pi$) is implicitly expressed in radians? Commented Jan 19, 2016 at 15:28
• I'm plotting it using Excel (for everyone else's convenience) and its trig functions work with radians. I just did a few tests and it does seem to have rounding errors though: $SIN(PI())$ gives $1.22515E-16$ ... Commented Jan 19, 2016 at 15:39
• Reading more closely: If $r = a\theta$, the spacing will be $2\pi a$, not $a$. But perhaps more importantly, something is odd with the way you've expressed $\theta$ (a variable angle) in terms of $w$ and $a$ (fixed dimensions). Commented Jan 19, 2016 at 15:54
• Thank you! My mistake was making $a$ equal to the spacing value, rather than as you suggested calculating a for the difference between one turn and the next ($\theta = 2\pi$). If you want to post that as an answer I'll be happy to accept it. Commented Jan 19, 2016 at 16:13

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*}