周 联, 王国瑾. 有理三角曲面的分片线性逼近[J]. 计算机研究与发展, 2012, 49(5): 1116-1122.
 引用本文: 周 联, 王国瑾. 有理三角曲面的分片线性逼近[J]. 计算机研究与发展, 2012, 49(5): 1116-1122.
Zhou Lian, Wang Guojin. Piecewise Linear Approximation of Rational Triangular Surfaces[J]. Journal of Computer Research and Development, 2012, 49(5): 1116-1122.
 Citation: Zhou Lian, Wang Guojin. Piecewise Linear Approximation of Rational Triangular Surfaces[J]. Journal of Computer Research and Development, 2012, 49(5): 1116-1122.

## Piecewise Linear Approximation of Rational Triangular Surfaces

• 摘要: 有理三角曲面的分片线性逼近在参数曲面的求交、绘制等方面有着重要应用.已有研究主要采用曲面的二阶导矢界来估计逼近误差, 而有理曲面的导矢界估计是一项困难的工作.为解决上述问题, 利用齐次坐标, 给出了一种定义域为任意三角形的有理三角曲面的分片线性逼近算法.该算法有效地避免了有理三角曲面的导矢界估计, 并且离散段数可先验地给出.此外, 通过重新参数化技术来缩小有理三角Bézier曲面的权因子之间的比值, 进一步提高了算法的效率.

Abstract: Piecewise linear approximation of rational triangular surfaces is useful in surfaces intersection, surfaces rendering and mesh generation. The approximation error bound is usually estimated based on the information about second-order derivative bounds of the rational triangular surfaces. But the derivative bounds of rational triangular surfaces are difficult and less effective to be estimated. To solve this problem, using homogeneous coordinates and inequality method, we present an algorithm to estimate subdivision depths for rational triangular surfaces which are defined in any arbitrary triangle. The estimation is performed on the polynomial surfaces, of which the given rational surfaces are the images under the standard perspective projection. It is more efficient than evaluating the derivative bounds of the given surfaces directly. The subdivison depth is obtained in advance, however, it guarantees the required flatness of the given surface after the subdivision. Moreover, using Mbius reparameterization technique, the variance of the log weights of rational triangular Bézier surfaces is minimized, which can obviously improve the efficiency of the algorithm. In particular, the optimal reparameterization is solved explicitily, so reparameterization hardly increases operating times. Numerical examples suggest that this algorithm not only possesses more powerful properties, but also is more effective compared with any other old methods.

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