Sets of straight line segments with special structures and properties appear in various applications of geometric modeling, such as scientific visualization, computer-aided design, and medical image processing. In this paper, we derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. In particular, we consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets.

中文翻译：

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线段集合中的交叉点和电路

具有特殊结构和特性的直线段集出现在几何建模的各种应用中，例如科学可视化，计算机辅助设计和医学图像处理。在本文中，我们以线段数为依据，推导了在具有一定结构的线段集合中可能出现的交点和封闭区域的数量的上界和下界。特别地，我们考虑其基础平面图为Halin图，仙人掌图，最大平面图和无三角形平面图的线段集，以及随机产生的线段集。