We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep \math generated by a moving circle. Given two ringed surfaces \math and \math , we formulate the condition \math (i.e. that the intersection of the two circles \math and \math is non-empty) as a bivariate equation \math of relatively low degree. Except for some redundant solutions and degenerate cases, there is a rational map from each solution of \math to the intersection point \math . Thus it is trivial to construct the intersection curve once we have computed the zero-set of \math. We also analyze some exceptional cases and consider how to construct the corresponding intersection curves.