Abstract
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.
Highlights
Multivariate splines are used extensively for computer-aided design (CAD) and, more recently, for computer-aided engineering (CAE)
The ideas we present here build upon those from [2], in multiple directions, and their presentation is motivated by our primary objectives: self-contained, explicit, non-uniform rational B-splines (NURBS)-compatible descriptions that can be and efficiently implemented within existing CAD software
We describe the usage of classical univariate NURBS to assemble C 1 rational multi-degree spline basis functions using an extraction matrix
Summary
Splines: Application to Construction of Exact Ellipses and Ellipsoids. Important note To cite this publication, please use the final published version (if applicable). Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10
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