Method and device for autonomous driving of a vehicle on a roadway in a direction of travel. A trajectory for driving on the roadway in the direction of travel is determined. A bending strip of limited length defines the trajectory, wherein the bending strip is fixed at one end thereof in a node which defines a starting point of the trajectory. A course of the trajectory is determined, starting from the starting point, in dependence on a bending line of the bending strip, which line extends, starting from the node, to the other end of the bending strip. A representation of a roadway boundary defines a boundary condition for the determination of the trajectory. A quality measure is defined in dependence on a property of the bending strip. The bending line which satisfies the boundary condition and for which the quality measure has an extremal value is determined.