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Summary form only given. A polygonal complex is a polygonal mesh that defines a curve with additional differential information such as tangent plane or normal and curvature values. In this sense, a polygonal complex corresponds to a curve interpolated by the limit surface of any polygonal mesh embodying it. We advance an approach for the deformation of subdivision surfaces under interpolation constraints. This is achieved by allowing the user to tag a configuration consisting of points, points with normal, or even control polygons and to deform the surface while maintaining the interpolation constraints. The constraints information can be converted, by means of a graphical user interface, into scalars defining various transformation parameters which have the ability to deform the surface when applied to the faces of the complex.