Continuous 2-dimensional space is often discretized by considering a mesh of weighted cells. In this work we study how well a weighted mesh approximates the space, with respect to shortest paths. We consider a shortest path $ \mathit{SP_w}(s,t) $ from $ s $ to $ t $ in the continuous 2-dimensional space, a shortest vertex path $ \mathit{SVP_w}(s,t) $ (or any-angle path), which is a shortest path where the vertices of the path are vertices of the mesh, and a shortest grid path $ \mathit{SGP_w}(s,t) $, which is a shortest path in a graph associated to the weighted mesh. We provide upper and lower bounds on the ratios $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert} $ in square and hexagonal meshes, extending previous results for triangular grids. These ratios determine the effectiveness of existing algorithms that compute shortest paths on the graphs obtained from the grids. Our main results are that the ratio $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $ is at most $ \frac{2}{\sqrt{2+\sqrt{2}}} \approx 1.08 $ and $ \frac{2}{\sqrt{2+\sqrt{3}}} \approx 1.04 $ in a square and a hexagonal mesh, respectively.

翻译：连续二维空间通常通过考虑由加权单元构成的网格进行离散化。本研究探讨加权网格在最短路径维度上对连续空间的逼近程度。我们考虑连续二维空间中从点$ s $到点$ t $的最短路径$ \mathit{SP_w}(s,t) $、顶点路径（任意角度路径）$ \mathit{SVP_w}(s,t) $（路径顶点均为网格顶点）以及网格图最短路径$ \mathit{SGP_w}(s,t) $（基于加权网格关联图的最短路径）。针对正方形网格与六边形网格，我们给出了比率$ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $、$ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $、$ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert} $的上下界，该工作将已有三角形网格结论进行了推广。这些比率决定了现有算法在网格衍生图上计算最短路径的有效性。核心结果包括：在正方形网格与六边形网格中，比率$ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $的上界分别为$ \frac{2}{\sqrt{2+\sqrt{2}}} \approx 1.08 $和$ \frac{2}{\sqrt{2+\sqrt{3}}} \approx 1.04 $。