This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is obeyed: any two adjacent curves do not cross, or any two non-adjacent curves cross at most once. Here, two curves are called adjacent if they share an endpoint. On the other hand, we demonstrate how to draw all curves such that any two adjacent curves cross exactly once, any two non-adjacent curves cross at least once and at most twice, and thus no two curves are disjoint. Furthermore, we analyze the emergence of disjoint curves without these mild assumptions, and characterize the plane structures in complete graph drawings guaranteed by each of the rules above.

翻译：本文探讨在纸上连接各点对并绘制曲线的问题。在温和假设下，我们证明若遵循以下任一规则，则许多两两不相交的曲线不可避免：任意两条相邻曲线不相交，或任意两条非相邻曲线至多相交一次。此处两条曲线若共享端点则称为相邻。另一方面，我们展示了如何绘制所有曲线，使得任意两条相邻曲线恰好相交一次，任意两条非相邻曲线至少相交一次且至多两次，从而不存在两条不相交的曲线。此外，我们分析了在没有这些温和假设时不相交曲线的出现情况，并刻画了上述每条规则所保证的完全图绘制中的平面结构。