Abstract: In this paper, we propose a new shape-modeling paradigm based on the concept of Lagrangian surface flow. Given an input polygonal model, the user interactively defines a distance field around regions of interest; the locally or globally affected regions will then automatically deform according to the user-defined distance field. During the deformation process, the model can always maintain its regularity and can properly modify its topology by topology merging when collisions between two different parts of the model occur. Comparing with level-set based methods, our algorithm allows the user to work directly on existing polygonal models without any intermediate model conversion. Besides closed polygonal models, our algorithm also works for mesh models with open boundaries. Within our framework, we developed a number of shape-modeling operators including blending, cutting, drilling, free-hand sketching, and mesh warping. We applied our algorithm to a variety of examples that demonstrate the usefulness and efficacy of the new technique in interactive shape design and surface deformation.