Unstructured triangular meshes are the most standard representation for surface geometry in computer graphics applications due to their simplicity and flexibility. Especially, meshes with special regular connectivity such as subdivision connectivity and with uniform spatial distribution are more useful for modeling, compression and so on. This paper proposes an efficient framework for interactively generating meshes with regular connectivity from those with arbitrary connectivity. The basic idea is to draw a regular mesh pattern into the two-dimensional embedded image of a part of a mesh using parameterization technique, partitioning a mesh to several disk-like regions by the user enables the control of geometric and connective features of a new mesh. One merit is that the user can select from various types of regular mesh patterns (lattice, radial etc.) suitable for such features and not only from the subdivision connectivity as seen in some recent researches about automatic remeshing. We also propose a fast remeshing algorithm endurable for the interactive operation. It rearranges vertices of a regular mesh pattern drawn in R2 so that the corresponding vertices in R3 are uniformly distributed on the original mesh. The algorithm is independent of the various mesh patterns and it is possible to use the results of the rearrangement as a template for the homogeneous re-parameterization. We demonstrate several remeshing examples with different resolutions.