A General Class of C¹ Smooth Rational Splines

A General Class of C¹ Smooth Rational Splines: Application to Construction of Exact Ellipses and Ellipsoids

Speleers, Hendrik (University of Rome Tor Vergata)
Toshniwal, D. (TU Delft Delft Institute of Applied Mathematics; TU Delft Numerical Analysis)

Delft Institute of Applied Mathematics

2021

In this paper, we describe a general class of C¹ smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids — some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design-through-analysis compatible extraction matrices; different sets of NURBS are allowed to have different polynomial degrees and weight functions. Tensor products of the univariate splines yield multivariate splines. In the bivariate setting, we describe how similar design-through-analysis compatible transformations of the tensor-product splines enable the construction of smooth surfaces containing one or two polar singularities. The material is self-contained, and is presented such that all tools can be easily implemented by CAD or CAE practitioners within existing software that support NURBS. To this end, we explicitly present the matrices (a) that describe our splines in terms of NURBS, and (b) that help refine the splines by performing (local) degree elevation and knot insertion. Finally, all C¹ spline constructions yield spline basis functions that are locally supported and form a convex partition of unity.

Exact ellipses and ellipsoids
Piecewise-NURBS representations
Smooth parameterizations

http://resolver.tudelft.nl/uuid:92180757-f15e-4709-b129-88abe71798b1

https://doi.org/10.1016/j.cad.2020.102982

0010-4485

Computer-Aided Design, 132

Institutional Repository

journal article

© 2021 Hendrik Speleers, D. Toshniwal