Software-engineering approach to degree elevation of B-spline curves

doi:10.1016/0010-4485(94)90004-3

Computer-Aided Design

Volume 26, Issue 1, January 1994, Pages 17-28

https://doi.org/10.1016/0010-4485(94)90004-3 Get rights and content

Abstract

A software-engineering approach to the degree elevation of B-spline curves is presented. A general method is introduced that consists of the following steps: (a) decompose the B-spline curve into piecewise Bézier curves by a modified version of knot insertion, (b) perform the necessary operation (degree elevation) on each Bézier segment, and (c) remove unnecessary knots. This general approach is applied to degree elevation, and it is shown that the resulting algorithm is very competitive with existing methods in speed, data storage, numerical accuracy and growth rate.

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Theorem 1 will be used in the proof of Theorem 2. The degree elevation algorithm of NURBS curves is studied in [22–25]. In 2007, Wang and Deng proved that the degree elevation of B-spline curve is the corner cutting procedure of its control polygon (Theorem 4, Section 4, [26]).
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In this paper, we introduce hybrid-degree weighted T-splines by means of local p-refinement and apply them to isogeometric analysis. Standard T-splines enable local h-refinement, however, they do not support local p-refinement. To increase the flexibility of T-splines so that local p-refinement is also available, we first define weighted B-spline curves of hybrid degree. Then we extend the idea to weighted T-spline surfaces of hybrid degree. A transition region is defined so as to stitch the locally p-refined region with the rest of the mesh. The transition region has the same basis function degree as the p-refined region, and the same surface continuity as the rest of the mesh. Finally, we compare the performance of odd-, even-, and hybrid-degree T-splines over an $L$ -shaped domain governed by the Laplace equation and create high-genus surfaces using hybrid-degree weighted T-splines.
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Citation Excerpt :
The degree elevation algorithm is also an important algorithm of NURBS curves. There are several methods studied on the degree elevation of B-spline curves, such as Refs. [26–29]. In 2007, Wang and Deng proved that the degree elevation of B-spline curve is the corner cutting procedure of its control polygon (Theorem 4, Section 4, [30]).

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^∗∗: Les A Piegle is a professor of computer science and engineering at the University of South Florida, USA. His research interests are in CAD/CAM, geometric modelling, user-interface design, data structures and algorithms, and computer graphics. He spent many years researching and implementing NURBS routines in academia as well as in industry.

^†: Wayne Tiller is an independent consultant specializing in geometric modelling and computational geometry for CAD/CAM. He has 21 years experience in applied mathematics, computer science, and software development. From 1981 to 1990, he worked at Structural Dynamics Research Corporation, USA, conducting research and developing software for their NURBS-based geometry products. He served on the IGES Geometry Sub-committee from 1981 to 1984. He has taught courses and conducted seminars in computational geometry at the University of Cincinnati, USA, the University of Texas at Tyler, USA, and in government and industry. He received a BS in mathematics in 1968 from Louisiana State University at Baton Rouge, USA, and a PhD in mathematics in 1972 from Texas Christian University in Forth Worth, USA.

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Software-engineering approach to degree elevation of B-spline curves

Abstract

Comput. Aided Geom. Des.

Comput. Aided Des.

Comput.-Aided Des.

Algorithms for degree-raising of splines

ACM Trans. Graph.