Finding the exact spanning ratio of a Delaunay graph has been one of the longstanding open problems in Computational Geometry. Currently there are only four convex shapes for which the exact spanning ratio of their Delaunay graph is known: the equilateral triangle, the square, the regular hexagon and the rectangle. In this paper, we show the exact spanning ratio of the parallelogram Delaunay graph, making the parallelogram the fifth convex shape for which an exact bound is known. The worst-case spanning ratio is exactly $$\frac{\sqrt{2}\sqrt{1+A^2+2A\cos(\theta_0)+(A+\cos(\theta_0))\sqrt{1+A^2+2A\cos(\theta_0)}}}{\sin(\theta_0)} .$$ where $A$ is the aspect ratio and $\theta_0$ is the non-obtuse angle of the parallelogram. Moreover, we show how to construct a parallelogram Delaunay graph whose spanning ratio matches the above mentioned spanning ratio.

翻译：确定Delaunay图的精确伸展比一直是计算几何领域中长期悬而未决的问题之一。目前仅已知四种凸形状的Delaunay图具有精确伸展比：等边三角形、正方形、正六边形和矩形。本文证明了平行四边形Delaunay图的精确伸展比，使平行四边形成为第五种已知精确界的凸形状。其最坏情况下的伸展比精确为$$\frac{\sqrt{2}\sqrt{1+A^2+2A\cos(\theta_0)+(A+\cos(\theta_0))\sqrt{1+A^2+2A\cos(\theta_0)}}}{\sin(\theta_0)}$$，其中$A$为纵横比，$\theta_0$为平行四边形的非钝角。此外，我们展示了如何构造一个伸展比与上述值完全匹配的平行四边形Delaunay图。