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Planning trajectories for nonholonomic systems is difficult and computationally expensive. When facing unexpected events, it may therefore be preferable to deform in some way the initially planned trajectory rather than to re-plan entirely a new one. We suggest here a method based on affine transformations to make such deformations. This method is exact and fast: the deformations and the resulting trajectories can be computed algebraically, in one step, and without any trajectory re-integration. To demonstrate the possibilities offered by this new method, we use it to derive position correction, orientation correction and obstacle avoidance algorithms for three classical nonholonomic systems: the unicycle, the bicycle, and an underwater vehicle.