We present sweeping line graphs, a generalization of $\Theta$-graphs. We show that these graphs are spanners of the complete graph, as well as of the visibility graph when line segment constraints or polygonal obstacles are considered. Our proofs use general inductive arguments to make the step to the constrained setting that could apply to other spanner constructions in the unconstrained setting, removing the need to find separate proofs that they are spanning in the constrained and polygonal obstacle settings.

翻译：我们展示了扫描线图,这是$\Theta$-graphs 的概括。 我们显示这些图表是完整图的射线符,以及考虑线段限制或多边形障碍时的可见度图。 我们的证明用一般的推论来向限制环境迈出一步,该限制环境可以适用于未受限制环境中的其他发射线的构造,从而消除了寻找单独的证据证明这些图形分布在受限制和多边形障碍环境中的必要性。