A General Class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e513" altimg="si128.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> Smooth Rational Splines: Application to Construction of Exact Ellipses and Ellipsoids

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