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

Hendrik Speleers, Deepesh Toshniwal*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)
30 Downloads (Pure)


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.

Original languageEnglish
Article number102982
Number of pages14
JournalCAD Computer Aided Design
Publication statusPublished - 2021


  • Exact ellipses and ellipsoids
  • Piecewise-NURBS representations
  • Smooth parameterizations


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