Bernstein Bezoutians and application to intersection problems

We study the Bézier curve–surface and Bézier surface–surface intersection problems avoiding the well-known unstable conversion between the Bernstein basis and the power basis. These varieties are given by parameterizations in Bernstein bases and all intermediate computations are performed in that form. For this purpose we construct an adapted resultant for generic Bernstein polynomial systems with a special shape which appear in the intersection problems. This construction is based on the expression of the Bezoutian matrix in Bernstein form.

► We study intersection problems in Bernstein bases using an algebraic approach.
► We construct a resultant for Bernstein polynomial systems based on a Bezoutian matrix.
► This matrix has a nice structure inherited from the special shape of these systems.
► Intersection problems are reduced to the resolution of structured linear systems.