Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values
Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values
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DOI：
10.1145/3596711.3596728
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发表时间：
2023-08
期刊：
影响因子：
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通讯作者：
J. Stam
中科院分区：
文献类型：
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作者：
J. Stam
在本文中，我们反驳的信念，广泛的计算机图形学社区的Catmull-Clark细分曲面不能直接评估没有明确细分。我们表明，表面及其所有衍生物可以评估的一组特征基函数，只依赖于细分方案，我们推导出这些基础函数的解析表达式。特别地，在Catmull-Clark曲面是双三次B样条的控制网格的规则部分上，特征基等于幂基。此外，我们的技术是易于实现和有效的。我们已经使用我们的实现来计算高质量的细分曲面的曲率图。我们的评估方案的成本是可比的双三次样条。因此，我们的方法允许许多算法开发的参数曲面适用于Catmull-Clark细分曲面。这使得细分曲面成为自由曲面建模的更具吸引力的工具。
In this paper we disprove the belief widespread within the computer graphics community that Catmull-Clark subdivision surfaces cannot be evaluated directly without explicitly subdividing. We show that the surface and all its derivatives can be evaluated in terms of a set of eigenbasis functions which depend only on the subdivision scheme and we derive analytical expressions for these basis functions. In particular, on the regular part of the control mesh where Catmull-Clark surfaces are bi-cubic B-splines, the eigenbasis is equal to the power basis. Also, our technique is both easy to implement and efficient. We have used our implementation to compute high quality curvature plots of subdivision surfaces. The cost of our evaluation scheme is comparable to that of a bi-cubic spline. Therefore, our method allows many algorithms developed for parametric surfaces to be applied to Catmull-Clark subdivision surfaces. This makes subdivision surfaces an even more attractive tool for free-form surface modeling.