中文摘要：

我们以取决于分面具的起始的控制点序列和常数的最大的差别在 k 褶层分以后估计在第三的分弄弯 / 出现和他们的控制多角形之间的错误界限。界限独立于分的过程并且没有递归的分，能被评估。我们的技术因此独立于 parametrization 它能容易并且高效地被实现。这为有用、重要预先计算错误在象表面 / 表面交叉那样的许多设计应用预先分弄弯 / 出现跳，网孔产生，用机器制造的 NC，等等显示的表面。数学题目分类：65D17， 65D07， 65D05。

英文摘要：

We estimate error bounds between ternary subdivision curves/surfaces and their control polygons after k-fold subdivision in terms of the maximal differences of the initial control point sequences and constants that depend on the subdivision mask. The bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Our technique is independent of parametrization therefore it can be easily and efficiently implemented. This is useful and important for pre-computing the error bounds of subdivision curves/surfaces in advance in many engineering applications such as surface/surface intersection, mesh generation, NC machining, surface rendering and so on.