TY - JOUR
T1 - A General Class of C1 Smooth Rational Splines
T2 - Application to Construction of Exact Ellipses and Ellipsoids
AU - Speleers, Hendrik
AU - Toshniwal, Deepesh
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - Exact ellipses and ellipsoids
KW - Piecewise-NURBS representations
KW - Smooth parameterizations
UR - http://www.scopus.com/inward/record.url?scp=85098461566&partnerID=8YFLogxK
U2 - 10.1016/j.cad.2020.102982
DO - 10.1016/j.cad.2020.102982
M3 - Article
AN - SCOPUS:85098461566
VL - 132
JO - Computer-Aided Design
JF - Computer-Aided Design
SN - 0010-4485
M1 - 102982
ER -