The Viro method for construction of Bernstein-Bézier algebraic hypersurface piece

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

“…Q k is preferred for AGCD computations because its condition number is smaller than the condition numbers of the other matrices in (10). The computation of its entries requires, however, the evaluation of three combinatorial terms, which is greater than the cost of the evaluation of the entries of the other matrices in (10).…”

“…Resultant matrices are frequently used for this computation, and these matrices and other polynomial computations also occur in robotics [5], computer vision [6], computational geometry, for example, the implicitization of parametric curves and surfaces [9] and the construction of surfaces [10,11], control theory [13] and the computation of multiple roots of a polynomial [17,22]. There are several resultant matrices, including the Sylvester, Bézout and companion resultant matrices, of which the Sylvester matrix is the most popular, presumably because its entries are linear, even though it is larger than the Bézout and companion matrices.…”

The Viro Method for Construction of Piecewise Algebraic Hypersurfaces

“…This section reviews briefly the Viro method for the construction of Bernstein-Bézier algebraic hypersurface piece, as stated in [18]. Throughout this paper, we denote by R + (resp., R * + ) the set of real numbers such that ≥ 0 (resp., > 0) and by Z + the set of nonnegative integers.…”

“…Moreover, the surface can be represented in Bernstein-Bézier form since it is often defined on a simplex, and writing a polynomial in it's the Bernstein-Bézier representation has significant advantages since its coefficients reflect geometric information about the shape of the polynomial surface, and the barycentric coordinates relative to the simplex are affine invariant, and Bernstein-Bézier basis polynomials exhibit many important properties (see [3,[15][16][17]). Therefore, based on the Viro method and the Newton polyhedra of Bernstein-Bézier polynomial, Lai et al in [18] established a new method for the construction of Bernstein-Bézier algebraic hypersurfaces on a simplex with a prescribed topology and presented a method to describe the topology of the Viro Bernstein-Bézier algebraic hypersurface piece.…”

“…In CAGD and geometric modelling, most of the complex curves and surfaces are expressed by piecewise polynomials with certain smoothness (see [3,17,18]). Thus, the aim of this paper is to establish a new method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology.…”