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：在计算几何领域，确定Delaunay图的精确跨度比一直是长期未解的开放问题之一。目前仅有四种凸形状的Delaunay图已知其精确跨度比：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图的精确跨度比，使平行四边形成为第五种已知精确界的凸形状。最坏情况下的跨度比精确为$$\frac{\sqrt{2}\sqrt{1+A^2+2A\cos(\theta_0)+(A+\cos(\theta_0))\sqrt{1+A^2+2A\cos(\theta_0)}}}{\sin(\theta_0)} .$$其中$A$是纵横比，$\theta_0$是平行四边形的非钝角。此外，我们还展示了如何构造一个跨度比与上述值相匹配的平行四边形Delaunay图。