Bernstein–Bezoutian matrices and curve implicitization

A new application of Bernstein–Bezoutian matrices, a type of resultant matrices constructed when the polynomials are given in the Bernstein basis, is presented. In particular, the approach to curve implicitization through Sylvester and Bézout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational parametric equations of the curve are expressed in the Bernstein basis, avoiding the basis conversion from the Bernstein to the power basis. The coefficients of the implicit equation are computed in the bivariate tensor-product Bernstein basis, and their computation involves the bidiagonal factorization of the inverses of certain totally positive matrices.