Abstract: This paper presents a framework for generating smooth-looking transformations between pairs of surfaces that may differ in topology. The user controls the transformation by specifying a sparse control mesh on each surface and by associating each face in one control mesh with a corresponding face in the other. The algorithm builds a transformation from this information in two steps. The first step constructs a series of shapes and meshes (according the theory of topological surgery) that describes how topological changes should occur at critical points during the transformation. This makes possible the second step, which establishes smooth transformations by combining intermediate shapes in this series. Control meshes allow the user precise but intuitive control of the morph, while the 3D surfaces that result can be used for rendering or keyframing.