Elsevier

Computer-Aided Design

Volume 33, Issue 13, November 2001, Pages 967-974
Computer-Aided Design

Choosing nodes and knots in closed B-spline curve interpolation to point data

https://doi.org/10.1016/S0010-4485(00)00133-0Get rights and content

Abstract

Node and knot placement for closed B-spline curve interpolation to point data heavily depends on whether the degree of B-splines is odd or not. For odd degree B-splines, the natural method of setting knots to coincide with nodes (i.e. parameters) works very well and provides the good quality. However, when the degree is even, the usual methods including the natural method can have problems and result in the bad quality. This paper presents a method, called the shifting method, which works well for even degree B-spline interpolation. It has nearly the same properties as the natural method does for odd degree B-splines. It is simple and provides the good quality of a resultant curve.

Section snippets

Problem definition

We assume that the reader is familiar with the concepts of B-spline curves. This paper considers the problem of closed B-spline curve interpolation to a sequence of points Pi (i=0,…,m) of a closed polygon, where (p−1) constraints are additionally imposed to satisfy C(p−2) continuity at the endpoints of a resulting curve C(t) of the pth order of B-splines. It is assumed that no two consecutive vertices on the polygon are the same.

The problem is basically the one of solving a system of linear

Previous methods for node and knot placement

Mostly the parameters t̄i at the points Pi can be given by the general exponent method [8]:t̄i=j=0i−1P(j+1)mod(m+1)Pjej=0mP(j+1)mod(m+1)Pje(i=1,…,m)where t̄0=0 and t̄m+1=1. It reduces to the uniform method with e=0, to the chord length method with e=1, and to the centripetal method with e=1/2. For odd degree B-splines, there are broadly three methods for knot placement: uniform, averaging, and natural methods. They all make it possible to construct a curve composed of (m+1) curve segments

Shifting method

Presented is a method, called the shifting method, of node and knot placement which works well for even degree B-spline interpolation. It starts with parameters computed via any parameterization method like the general exponent method [8]. Let the ith parameter span di be the difference from the (i+1)th parameter t̄i+1 to the ith parameter t̄i, that is, di=t̄i+1t̄i, (i=0,…m). It then defines domain knots ξi such that, for each parameter t̄i, a knot span is built by adding halves of two

Examples

A cross-shaped polygon shown in Fig. 2is used to demonstrate the quality of the shifting method. It consists of 12 points ordered in a clockwise direction. The starting point is denoted by the hollow rectangle. Cyclic knot vectors are used for the convenience of showing the symmetry of control polygons and that of resultant closed B-spline curves. Note that the shifting method works with clamped knot vectors as well as cyclic knot vectors. It is just domain knots that are meaningful in the

Conclusion

In the problem of closed B-spline curve interpolation to a sequence of points of a closed polygon, node and knot placement heavily depends on whether the degree of B-splines is odd or not. As the natural method works well for odd degree B-splines, the shifting method presented in this paper works well for even degree B-splines. It makes the system matrix well-conditioned and provides the good quality of a closed B-spline curve. It is very simple, and it does not need to increase the number of

Acknowledgements

This study was supported in part by research funds from Chosun University, 2001.

Hyungjun Park is a full-time professor in the Department of Industrial Engineering at Chosun University, Korea. He received a BS, an MS, and a PhD in industrial engineering from Pohang University of Science and Technology, Korea, in 1991, 1993, and 1996, respectively. From 1996 to 2001, he worked as a senior researcher at Samsung Electronics, Korea. He was involved in developing a commercial CAD/CAM software VX Vision and an in-house software for modeling and manufacturing aspheric lenses used

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Hyungjun Park is a full-time professor in the Department of Industrial Engineering at Chosun University, Korea. He received a BS, an MS, and a PhD in industrial engineering from Pohang University of Science and Technology, Korea, in 1991, 1993, and 1996, respectively. From 1996 to 2001, he worked as a senior researcher at Samsung Electronics, Korea. He was involved in developing a commercial CAD/CAM software VX Vision and an in-house software for modeling and manufacturing aspheric lenses used in various optical products. His research interests include geometric modeling, reverse engineering for 3D shape construction, computational geometry, and software engineering for CAD/CAM systems.

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