Abstract: The quality of subdivision surface generated by the approximating scheme is usually better than that by the interpolating scheme, while the approximating subdivision surface is unable to interpolate the vertices of the initial control mesh. Traditional methods that make the approximating subdivision surface interpolate the initial mesh need to solve a global linear system. It is computation-intensive, and hard to deal with dense meshes. Without solving a linear system, the progressive interpolation calculates the approximating subdivision surface that interpolates the initial mesh by adjusting the vertices of the control mesh iteratively. It can handle control meshes of any size and any topology while generating smooth subdivision surfaces that faithfully resemble the shape of the initial meshes. In this paper, we show the local property of the progressive interpolation for approximating subdivision schemes. That is, if only a subset of the vertices of the control mesh are adjusted, and others remain unchanged, the limit of the subdivision surface generated in the progressive interpolation procedure still interpolates the corresponding subset of the vertices in the initial mesh. The local property of the progressive interpolation brings more flexibility for shape controlling, and makes the adaptive fitting possible. Lots of experimental examples illustrate the shape controlling and adaptive fitting capabilities of the local progressive interpolation.ss