In this paper, a theoretical characterisation of the topological errors which arise during the approximation of Euclidean distances from discrete ones is presented. The continuous distance considered is the widely used Euclidean distance whereas we consider as discrete distance the chamfer distance based on 3 x 3 masks. The objective is to obtain formal results from which algorithms for the exact solution of the Euclidean Distance Transformation using integer arithmetic can be derived. We conclude this study by presenting a global upper bound for a topologically-correct distance mapping, irrespective of the chamfer distance coefficients, and identify the smallest coefficients associated with this bound.