Estimating tessellation parameter intervals for rational curves and surfaces
Estimating tessellation parameter intervals for rational curves and surfaces
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DOI：
10.1145/343002.343034
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Jianmin Zheng;T. Sederberg
中科院分区：
文献类型：
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作者：
Jianmin Zheng;T. Sederberg
本文提出了一种先验地确定一个常参数区间的方法，用于对有理曲线或曲面进行镶嵌，使曲线或曲面与其分段线性逼近的偏差在规定的公差范围内。参数区间的估计是基于齐次坐标下二阶导数的信息，而不是直接使用仿射坐标。这个新的步长与Cheng[1992]的步长计算量大致相同，但可以证明它总是大于Cheng的步长。事实上，数值实验表明，新步长通常比Cheng[1992]的步长大几个数量级。此外，对于有理三次和四次曲线，新的步长通常是通过计算二阶导数函数的Bernstein多项式系数的边界得到的步长的两倍。
This paper presents a method for determining a priori a constant parameter interval for tessellating a rational curve or surface such that the deviation of the curve or surface from its piecewise linear approximation is within a specified tolerance. The parameter interval is estimated based on information about second-order derivatives in the homogeneous coordinates, instead of using affine coordinates directly. This new step size can be found with roughly the same amount of computation as the step size in Cheng [1992], though it can be proven to always be larger than Cheng's step size. In fact, numerical experiments show the new step is typically orders of magnitude larger than the step size in Cheng [1992]. Furthermore, for rational cubic and quartic curves, the new step size is generally twice as large as the step size found by computing bounds on the Bernstein polynomial coefficients of the second derivatives function.