Choosing nodes and knots in closed B-spline curve interpolation to point data
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 (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 at the points can be given by the general exponent method [8]:where and 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 to the ith parameter that is, (i=0,…m). It then defines domain knots ξi such that, for each parameter 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
References (10)
- et al.
Smooth surface approximation to serial cross-sections
Computer-Aided Design
(1996) - et al.
A method for approximate NURBS curve compatibility based on multiple curve refitting
Computer-Aided Design
(2000) Choosing nodes in parametric curve interpolation
Computer-Aided Design
(1989)A practical guide to splines
(1978)Curves and surfaces for computer aided geometric design: a practical guide
(1993)
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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.