A Note on Rational Lagrange Polynomials for CAGD Applications and Isogeometric Analysis

While the established theory of computer-aided geometric design (CAGD) suggests that rational Bernstein–Bézier polynomials associated with control points can be used to accurately represent conics and quadrics, this paper shows that the same goal can be achieved in a different manner. More specifically, rational Lagrange polynomials of the same degree, associated with nodal points lying on the true curve or surface, can be combined with appropriate weights to yield equivalent numerical results within a Bézier patch. The specific application of this equivalence to derive weights for Lagrange nodes on conics and quadrics is shown in this paper. Although this replacement may not be crucial for CAGD purposes, it proves useful for the direct implementation of boundary conditions in isogeometric analysis, since it allows the use of nodal values on the exact boundary.