Determination of the world surface of a relativistic string from the trajectory of a massive end

It is shown that the world surface of a relativistic string can be uniquely recovered from the trajectory of a massive end of it if the trajectory is specified in Minkowski space in the form of an arbitrary curve of constant curvature with timelike tangent vector. Conditions are determined for the existence on the world surface of a line that can be identified with the trajectory of the other end of the string and also for the possibility of physical realization of the model in the case when there is no such line.