Solving Bezout-like polynomial equations for the design of interpolatory subdivision schemes

Subdivision schemes are nowadays customary in curve and surface modeling. In this paper the problem of designing interpolatory subdivision schemes is considered. The idea is to modify a given approximating subdivision scheme just enough to satisfy the interpolation requirement. From an algebraic point of view this leads to the solution of a generalized Bezout polynomial equation possibly involving more than two polynomials. By exploiting the matrix counterpart of this equation it is shown that small-degree solutions can be generally found by inverting an associated structured matrix of Toeplitz-like form. If the approximating scheme is defined in terms of a free parameter, then the inversion can be performed by numeric-symbolic methods.