Survey on geometric iterative methods and their applications

Geometric iterative methods (GIM), including the progressive-iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, present the local properties and accelerating techniques, and show their convergence. Moreover, because it is easy to integrate geometric constraints in the iterative procedure, GIM has been widely applied in geometric design and related areas. We survey the successful applications of geometric iterative methods, including applications in geometric design, data fitting, reverse engineering, mesh and NURBS solid generation.