The calculation of intersection curves between (parametric) surfaces is one of the main tasks of Computer Aided Geometric Design (CAGD). There exist several classes of algorithms for solving this problem, among them {\it marching methods} in which we start with a point belonging to the intersection and try to follow the corresponding intersection curve point-by-point. The existing methods are usually approximate ones. These methods work more or less fine on each branch of the intersection of two surfaces, but when we approach the intersection of the branches, we often, due to the approximate character of the existing methods, jump to another branch instead of following the original one. As a result of such a jump, we get a wrong topological picture of the intersection.

The author proposes a new interval-based marching method that results in a {\it guaranteed} interval for the intersection curve and that avoids the jumps.

Since we need guaranteed interval computations, this method requires reasonably more computation time than the existing approximate marching techniques.