Generating surfaces by refinement of curves

We propose to extend the corner cutting subdivision schemes from refining control points to refining control curves. Such schemes are relevant to the design of a surface by a family of curves. Our schemes replace averages between two consecutive control points by similar weighted averages between two consecutive curves, based on a geometric correspondence between points on the two curves. We show that the schemes converge under mild conditions on the initial control curves, and derive properties of the limit.