Qualitative and quantitative comparisons of B-spline offset surface approximation methods
Section snippets
Surface offsetting
The first step in this research was to conduct a survey of known curve and surface offset methods.
A set of surface offset methods was compiled. (These methods will be discussed in detail in Section 4.) Some previously developed methods that were just for planar curves or just for cubic curves were extended to surfaces. Some of the studied methods preserve the original smoothness of the base surface, an important criterion in exterior surface modeling, and some do not. Below is a list of the
Surface definition
A tensor product B-spline surface has the form [1]:where fip(u),fjq(v) are univariate B-spline basis functions of degrees p, q, respectively, and Pi,j are control points in . The (u, v) domain of this mapping is a rectangle in general.
Offset error analysis method
In the first phase of the research, the offset surface obtained from each offset method was compared to the actual offset computed on a subset of points from the original domain, using an analytical definition of the surface offset. While the above method provides an approximation of the error, it involves a large number of computations, and may miss some surface features.
In this research we were looking for an error approximation scheme that would apply to Bézier and to B-spline surfaces. We
Surface offset methods
Most of the tested surface offset algorithms require that the original surface be decomposed to a set of Bézier patches. Since in most cases the algorithms perform the offsetting on each patch, it is essential to know if the surface offset method preserves surface continuity between patches. Some of the tested methods do preserve up to C2 (curvature) continuity. Other methods might preserve only C1 (tangent) continuity, or only C0 (positional) continuity.
The following methods will be discussed
Qualitative comparison of surface offsetting methods
In a qualitative comparison of offset methods, the objective is to provide information on distinguishing attributes of these methods, their characteristics, or possession of some special qualities. The following list is a summary of such a comparison.
- 1.
Geometric methods usually underestimate the offset operator. During our research, we found the approximate offset obtained using geometric methods for a surface consisting of elliptic points will lie in a space between the original surface and the
Quantitative comparison of surface offsetting methods
In quantitative comparison, the major criterion is how efficiently each method approximates the offset surface given a prescribed tolerance. We compared the number of control points with respect to the accuracy of offset approximation, and the required offset distance. The following list is a summary of such a comparison.
- 1.
Overall, numeric methods (especially least squares methods) reached a preset error tolerance faster than geometric methods.
- 2.
Overall, the best numeric surface offset methods are
Comparison data
The summary of comparison charts [7] was organized in two groups: results for strict conditions (offset distance equal to 1 unit, and offset tolerance equal to 0.0001 units), and results for ‘loose’ conditions (offset distance equal to 2.5, and offset tolerance equal to 0.1).
The resulting charts illustrate the overall performance of all methods for a given surface and given conditions. Summary test results showed how many control points were required to offset a surface using all researched
Conclusion and recommendation
Throughout the conducted tests, we have observed the following consistent results:
- 1.
The Tiller–Hanson–Nachman method performs well in cases of surfaces with only elliptic points. This observation applies to most of the methods. Only approximation methods seem to be resistant to cases of rapidly changing curvature of the boundary curves.
- 2.
The biggest advantage of the geometric method is that it guarantees the same level of internal parametric continuity as the original surface. However, to maintain
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