Skew Voronoi diagrams

O. Aichholzer, F. Aurenhammer, D. Chen, D. Lee, and E. Papadopoulou


On a tilted plane $T$ in three-space, skew distances are defined as the Euclidean distance plus a multiple of the signed difference in height. Skew distances may model realistic environments more closely than the Euclidean distance. Voronoi diagrams and related problems under this kind of distances are investigated. A relationship to convex distance functions and to Euclidean Voronoi diagrams for planar circles is shown, and is exploited for a geometric analysis and a plane-sweep construction of Voronoi diagrams on $T$. An output-sensitive algorithm running in time $O(n \log h)$ is developed, where $n$ and $h$ is the number of sites and non-empty Voronoi regions, respectively. The all nearest neighbors problem for skew distances, which has certain features different from its Euclidean counterpart, is solved in $O(n \log n)$ time.

Reference: O. Aichholzer, F. Aurenhammer, D. Chen, D. Lee, and E. Papadopoulou. Skew Voronoi diagrams. Int'l. Journal of Computational Geometry & Applications, 9:235-247, 1999.