Bernstein-Bezier Form Research Articles

Interactive deformable models with quadratic bases in Bernstein–Bézier-form

We present a physically based interactive simulation technique for de formable objects. Our method models the geometry as well as the displacements using quadratic basis functions in Bernstein–Bezier form on a tetrahedral finite element mesh. The Bernstein–Bezier formulation yields significant advantages compared to approaches using the monomial form. The implementation is simplified, as spatial derivatives and integrals of the displacement field are obtained analytically avoiding the need for numerical evaluations of the elements’ stiffness matrices. We introduce a novel traversal accounting for adjacency in order to accelerate the reconstruction of the global matrices. We show that our proposed method can compensate the additional effort introduced by the co-rotational formulation to a large extent. We validate our approach on several models and demonstrate new levels of accuracy and performance in comparison to current state-of-the-art.

A multivariate Powell–Sabin interpolant

We consider the problem of constructing a C 1 piecewise quadratic inter- polant, Q, to positional and gradient data defined at the vertices of a tessellation of n-simplices in IR n . The key to the interpolation scheme is to appropriately subdivide each simplex to ensure that certain necessary geometric constraints are satisfied by the subdivision points. We establish these constraints using the Bernstein-Bezier form for polynomials defined over simplices, and show how they can be satisfied. When constructed, the interpolant Q has full approximation power.

Tight linear envelopes for splines

A sharp bound on the distance between a spline and its B-spline control polygon is derived. The bound yields a piecewise linear envelope enclosing spline and polygon. This envelope is particularly simple for uniform splines and splines in Bernstein-Bezier form and shrinks by a factor of 4 for each uniform subdivision step. The envelope can be easily and efficiently implemented due to its explicit and constructive nature.