We initiate the study of spanners under the Hausdorff and Fr\'echet distances. We show that any $t$-spanner of a planar point-set $S$ is a $\frac{\sqrt{t^2-1}}{2}$-Hausdorff-spanner and a $\min\{\frac{t}{2},\frac{\sqrt{t^2-t}}{\sqrt{2}}\}$-Fr\'echet spanner. We also prove that for any $t > 1$, there exist a set of points $S$ and an $\varepsilon_1$-Hausdorff-spanner of $S$ and an $\varepsilon_2$-Fr\'echet-spanner of $S$, where $\varepsilon_1$ and $\varepsilon_2$ are constants, such that neither of them is a $t$-spanner.

翻译：本文首次在 Hausdorff 距离与 Fréchet 距离下开展 spanner 研究。我们证明：对于平面点集 $S$ 的任意 $t$-spanner，它同时是 $\frac{\sqrt{t^2-1}}{2}$-Hausdorff-spanner 和 $\min\{\frac{t}{2},\frac{\sqrt{t^2-t}}{\sqrt{2}}\}$-Fréchet-spanner。此外，我们证明：对于任意 $t > 1$，存在点集 $S$ 及其 $\varepsilon_1$-Hausdorff-spanner 和 $\varepsilon_2$-Fréchet-spanner（其中 $\varepsilon_1$ 和 $\varepsilon_2$ 为常数），使得两者均不是 $t$-spanner。