In this paper a method for fitting open surfaces to an unorganized set of data points is presented using a level set representation of the surface. This is done by tracking a curve, representing the boundary, on the implicitly defined surface. This curve is given as the intersection of the level set describing the surface and an auxiliary level set. These two level sets are propagated using the same motion vector field. Special care has to be taken in order for the surfaces not to intersect at other places than at the desired boundary. Methods for accomplishing this are presented and a novel fast scheme for finding good initial values is proposed. This novel method gives a piecewise linear approximation of the initial surface boundary using a partition of the convex hull. With the described method open surfaces can be fitted to point clouds obtained using structure from motion techniques. This paper solves an important practical problem since in many cases the surfaces in the scene are open or can only be viewed from certain directions. Successful experiments on several data sets support the method.