The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any $$\varepsilon >0$$ , the triangles can be cut into $$O(n^{3/2+\varepsilon })$$ connected semialgebraic pieces, whose description complexity depends only on the choice of $$\varepsilon $$ , such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. The pieces can be constructed efficiently. This work extends the recent study by two of the authors (Discrete Comput. Geom. 59(3), 725–741 (2018)) on eliminating depth cycles among lines in 3-space. Our approach is again algebraic, and makes use of a recent variant of the polynomial partitioning technique, due to Guth (Math. Proc. Camb. Philos. Soc. 159(3), 459–469 (2015)), which leads to a recursive algorithm for cutting the triangles. In contrast to the case of lines, our analysis here is considerably more involved, due to the two-dimensional nature of the objects being cut, so additional tools, from topology and algebra, need to be brought to bear. Our result makes significant progress towards resolving a decades-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics. In addition, we generalize our bound to well-behaved patches of two-dimensional algebraic surfaces of constant degree.

中文翻译：

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消除三维三角形之间的深度循环

3 空间中 n 对开放不相交三角形之间的垂直深度关系可能包含循环。我们证明，对于任何 $$\varepsilon >0$$ ，三角形可以被切割成 $$O(n^{3/2+\varepsilon })$$ 连接的半代数部分，其描述复杂度仅取决于选择$$\varepsilon $$ ，这样这些片段之间的深度关系现在是一个适当的偏序。在最坏的情况下，这个界限几乎是紧的。可以有效地构建这些部件。这项工作扩展了两位作者 (Discrete Comput. Geom. 59(3), 725–741 (2018)) 最近关于消除 3 空间中线之间的深度循环的研究。由于 Guth (Math. Proc. Camb. Philos. Soc. 159(3), 459–469 (2015))，我们的方法再次是代数的，并利用多项式分区技术的最新变体，这导致了用于切割三角形的递归算法。与线的情况相比，我们在这里的分析要复杂得多，因为被切割对象的二维性质，因此需要使用拓扑和代数的额外工具。我们的结果在解决计算几何中数十年的开放问题方面取得了重大进展，该问题由计算机图形学中的隐藏表面去除驱动。此外，我们将我们的界限推广到行为良好的常数度二维代数曲面片。我们的结果在解决计算几何中数十年的开放问题方面取得了重大进展，该问题由计算机图形学中的隐藏表面去除驱动。此外，我们将我们的界限推广到行为良好的常数度二维代数曲面片。我们的结果在解决计算几何中数十年的开放问题方面取得了重大进展，该问题由计算机图形学中的隐藏表面去除驱动。此外，我们将我们的界限推广到行为良好的常数度二维代数曲面片。