A method for deforming curves in a given image to a desired position in a second image is introduced in this paper. The algorithm is based on deforming the first image toward the second one via a Partial Differential Equation (PDE), while tracking the deformation of the curves of interest in the first image with an additional, coupled, PDE. The tracking is performed by projecting the velocities of the first equation into the second one. In contrast with previous PDE-based approaches, both the images and the curves on the frames/slices of interest are used for tracking. The technique can be applied to object tracking and sequential segmentation. The topology of the deforming curve can change without any special topology handling procedures added to the scheme. This permits, for example, the automatic tracking of scenes where, due to occlusions, the topology of the objects of interest changes from frame to frame. In addition, this work introduces the concept of projecting velocities to obtain systems of coupled PDEs for image analysis applications. We show examples for object tracking and segmentation of electronic microscopy. Index Terms: Partial differential equations, curve evolution, morphing, segmentation, tracking, topology.