The problem of obtaining smooth curves from a set of initial points through introducing new intermediate points is the topic of this paper. Such a technique of subdivision is widely used in modeling and rendering systems. According to the traditional four-point scheme of subdivision, the initial set of points S is extended iteratively, doubling at each step the number of points. Thus, a curve C is obtained.

The authors pose the question whether the initial set S can be interpolated, inserting at each step k new points between any two consecutive points of S, so that the generated curve is again C. They prove that this is only possible when the four-point scheme is applied repeatedly. Therefore, C is not invariant under different refinements of S, except the refinement by one or more steps of the 4-point scheme itself.

The paper may be of theoretical importance for researchers working on curve interpolation.