Fast and Local Fairing of B-Spline Curves and Surfaces
The paper proposes a fast fairing algorithm for curves and surfaces. It first defines a base algorithm for fairing curves, which is then extended to the surface case, where the isocurves of the surface are faired. The curve fairing process involves the di
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University of Tokyo Geomagic Hungary
Abstract. The paper proposes a fast fairing algorithm for curves and surfaces. It first defines a base algorithm for fairing curves, which is then extended to the surface case, where the isocurves of the surface are faired. The curve fairing process involves the discrete integration of a pseudo-arc-length parameterization of B-spline curves, with a blending and fitting phase concluding the algorithm. In the core of the fairing method, there is a fairness measure introduced in an earlier paper of the authors. This measure is based on the deviation from an ideal or target curvature. A target curvature is a series of smooth curvature values, generated from the original curve or surface. This curve and surface fairing technique is local and semi-automatic, but the user can also designate the region to be faired. The results are illustrated by a few examples on real-life models. Keywords: Curves and Surfaces, Geometric Optimization, Reverse Engineering.
1
Introduction
Fairing curves and surfaces plays an important role in CAGD, especially in the automobile industry. Connected curve nets and surfaces have to be smoothed while preserving their original features and connectivity. Automating this process is a crucial problem of Digital Shape Reconstruction1 [3]. Fairness does not have an exact mathematical definition, and it may have a different meaning depending on the context [9]. Still, there are some common properties of what we would call fair. It certainly includes some kind of mathematical continuity, e.g. C 2 or G2 . It also incorporates the requirement that a surface should have even reflections. Another important, although less intuitive, requirement is the smooth transition of curvature [6]. These properties can be tested using an arsenal of interrogation methods, e.g. isophote lines, curvature combs or curvature maps. But even with these, finding and mending small artifacts is a laborious manual process. The concept of target curvature and an iterative algorithm based on it is reviewed in Section 2. A new, fast algorithm is explained in Section 3, accompanied by test results in Section 4. 1
Formerly called Reverse Engineering.
F. Chen and B. Jüttler (Eds.): GMP 2008, LNCS 4975, pp. 155–163, 2008. c Springer-Verlag Berlin Heidelberg 2008
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P. Salvi, H. Suzuki, and T. Várady
(a) Curve with curvature comb
(b) The same curve with the target curvature comb.
Fig. 1. Target curvature of a cubic curve
2
Related Work
There is an abundant literature on creating fair curves and surfaces, dealing with both the definition of fairness and smoothing algorithms. Here we will only cover a previous publication of the authors, defining a fairness measure based on a target curvature. The algorithm in Section 3 will use this measure to generate fair curves and surfaces. This section also presents an iterative algorithm based on the same measure. For a more comprehensive review on fairness measures and algorithms, look at [9,10]. 2.1
Fairness Measure
Most fairing algorithms use some
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