2 circles of radius r_{1} and r_{2} are placed such their centers are D_{1}+D_{2} apart where D_{1} and D_{2} are the diameters of the circles. For ease of explanation lets assume circle one is centered at the origin and circle 2 is centered at (D_{1}+D_{2},0).

Now if we wrap a belt around the 2 circles so its perfectly tight, then L(D_{1},D_{2}) = length of the belt.
Its easy to see if D_{1}=D_{2} then the L(D,D) = pi*D+2*(D1+D2). The belt is tangent to each circle for 180 degrees, then the top and bottom lengths are lines going from (0,D) to (D1+D2,D) and (0,-D) to (D1+D2,-D).

Now it gets trickier if D1 not equal to D2. My instinct is that if D1 < D2, then the belt will be tangent to Circle_{1} for 180 - [theta_{1}] degrees; and will be tangent to Circle_{2} for 180+[theta_{2}] degrees. I also think that [theta_{1}] = k*[theta_{2}] where k is some function of D_{1} and D_{2}. But I have no idea about where to start to evaluate the thetas.

Draw in the radius for each circle which is perpendicular to the tangent, and join the centers of the circles with a line. You now have a trapezium (with two angles of 90 degrees), for which you can calculate the angles with trig.

By trapezium do you mean quadrilateral with no sides parallel or with 2 sides parallel?
Assuming you don't know if any sides are parallel. Let say Point A is the origin, Point B is the center of Circle_{2}, Point C is where the line is tangent to Circle_{1} and Point D is where the line is tangent to Circle_{2}.

We know that:
Angle ACD is 90
Angle BDC is 90

AB is length D1+D2
AC is length r_{1}
BD is length r_{2}.

Without knowing any of the angles is this enough to determine length of CD?

Last edited by drwhat on Tue Nov 08, 2011 3:49 am, edited 1 time in total.