Chapter 2Lagrange and Bézier Interpolants
Lagrange interpolation functions (interpolants) make it possible to construct geometric entities, edges and faces of finite elements or, simply, curves and surfaces as they are. Beyond this purely geometric aspect, these interpolants make it possible to construct approximations of "physical" functions (temperature, pressure, speed, etc.) involved in the problem under study.
In this chapter, we will show that the classic formulation using Lagrange interpolants and nodes is geometrically equivalent to a formulation in Bézier formalism, which is based on Bernstein polynomials and control points. We will also show that the two forms of writing are equivalent when we consider the approximation of a function defined via Lagrange interpolants and nodal values, or via Bernstein polynomials and control values.
We establish that the two systems of representation are analogous. We will show how to express a Lagrange function in Bézier formalism. Conversely, we will then show how to express a Bézier function in Lagrange formalism. By applying these results to curves and then patches themselves, we will show how, with the nodes being given, we can construct the corresponding control points, as well as how, with the control points being given, to construct the corresponding nodes. The case of classic (complete) patches and those of reduced patches are essentially detailed in two dimensions but the corresponding mechanism can be extended to three ...