A general class of $C^1$ smooth rational splines: Application to construction of exact ellipses and ellipsoids

TL;DR

This paper introduces a versatile class of $C^1$ smooth rational splines that precisely model ellipses and ellipsoids, facilitating their integration into CAD and CAE systems with explicit matrices for implementation.

Contribution

It presents a novel framework for constructing $C^1$ smooth rational splines with exact conic representations, compatible with existing NURBS-based CAD/CAE tools.

Findings

Enables exact modeling of ellipses and ellipsoids

Provides explicit matrices for spline construction and refinement

Supports local degree elevation and knot insertion

Abstract

In this paper, we describe a general class of $C^1$ 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 $C^1$ spline constructions yield spline basis functions that are locally supported and form a convex partition of unity.