Finding the distance between two circles in three-dimensional space

In this paper we investigate, from an algebraic point of view, the problem of finding the distance between two circles located in ℝ
³
. We show, by combining a theorem about solvable permutation groups and some explicit calculations with a computer algebra system, that, in general, the distance between two circles is an algebraic function of the parameters defining them, but that this function is not solvable in terms of radicals. Although this result implies that one cannot find a “closed-form” solution for the distance between an arbitrary pair of circles in ℝ
³
, we discuss how such an algebraic quantity can still be manipulated symbolically by combining standard polynomial operations with an algorithm for isolating the real roots of a polynomial in a convenient data structure for real algebraic numbers. This data structure and its operations have been implemented.