How to find distance between two different circles I am trying the find the distance between two different sized circles, both centred on the horizontal plane. I know the diameter of each circle, and the length around both circles if wrapped like a bike chain around the front and rear sprockets.
I've tried searching for the formula's but everything seems to work with a known distance to find the external tangent length's etc.
What are the steps to calculate this?
Thanks,
Matt
Update:
Found a couple of formulas from timing pulley sites that seem to do the job. Would still like to understand how they get built just out of interest.
Formula #1
$C = A + \sqrt{ A^2 + B }$
where
$A = \frac L4 - \pi \frac{D + d}{8}$
$B = \frac{\left(D - d\right)^2}{8}$

Formula #2
$C = \frac{A + \sqrt{A - 32 \left(D - d\right)^2} }{16}$
where
$A = 4L - 2\pi \left(D + d\right)$

Both come out with the same answer. Which isn't really a surprise as they look very similar.

Let $r, R$ be the radiuses, $L$ the length of chain, $D$ the distance and $\alpha$ the angle between the line joining centers and the radius of the bigger circle in radians. Then we have $$\frac L 2 = \alpha r + (\pi - \alpha)R + (R-r)\tan \alpha$$ and $$R-r = \cos \alpha D$$
These equations are sufficient to find $D$, but the calculation itself might be unpleasant.
