The problem of motion planning for two-degree-of-freedom systems is considered when obstacles to be avoided are bounded in configuration space by an arrangement of simple Jordan arcs. The existence of the continuous motion path connecting given initial and final configuration in a cluttered workspace is the issue, then the motion planning problem is considered. As a collision-free motion exists iff the initial and final configuration belongs to the same connected component of the so called free configuration space the attention is focussed on topological, combinatorial and algorithmic issues involving single face problem for an arrangement of Jordan curves. An upper bound for the time- and space- complexity of the algorithm is given in terms of the number of collision constraints and the maximum algebraic degree of these constraints (the algorithm is supposed to solve the problem of finding a collison-free path). Davenport-Schinzel sequences are used to establish the complexity measure. Although the upper bounds for the computations cost are not very restrictive given a 2 d.o.f. system and quite reallistically modelled obstacles boundaries, it is claimed that the extension of the approach to the 3 d.o.f. case is not straightforward nor simple. Included proofs are of constructive nature. The whole work can be viewed as the generalisation of the earlier studies single face problem for an arrangement of line segments.