Abstract In learning by demonstration, the generalization of motion trajectories far away from the set of demonstrations is often limited by the dependency of the learned models on arbitrary coordinate references. Trajectory shape descriptors have the potential to remove these dependencies by representing demonstrated trajectories in a coordinate-free way. This paper proposes a constraint-based optimization framework to generalize demonstrated rigid-body motion trajectories to new situations starting from the shape descriptor of the demonstration. Experimental results indicate excellent generalization capabilities showing how, starting from only a single demonstration, new trajectories are easily generalized to novel situations anywhere in task space, such as new initial or target positions and orientations, while preserving similarity with the demonstration. The results encourage the use of trajectory shape descriptors in learning by demonstration to reduce the number of required demonstrations.