Abstract: Planar shape deformation has wide applications in computer graphics.Traditional mesh-based methods can flexibly control the distortions of the deformation while has the weakness of non-smoothness due to its piecewise-linear nature.And the mesh-free methods have the feature of smoothness,but most of them are lack of distortion control.Though a few take distortions into consideration,the computation is complicated and time-consuming.To this end,we combine the advantages of both the two kinds of methods and propose a radial basis functions(RBF)-based planar shape deformation combining with mesh-based method.It constructs triangle-mesh structure for the rest shape domain and uses RBF to represent the deformation,and then builds the optimization equations for the deformation with locally injective constraints.To solve the problem efficiently,it borrows the computational framework of the mesh-based methods,which calculates geodesic fields for the radial basis functions and linearly approximates the numerical calculus of deformation energy by using the mesh structure.A large number of comparative experiments show that our deformation results are more smooth,natural and with lower distortions than the traditional mesh-based methods,and it is also more efficient than traditional mesh-free methods with comparative results.Meanwhile,the method has the feature of controllable,simple and extendable.