Skew Voronoi Diagrams

Abstract

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 analysisis 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 are the numbers 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.

Optimisation of the regional energy supply network: a multi-objective analysis in the province of Florence (Italy)

“…Following the notation of Renfrew and Level (1979) and Aichholzer et al (1999), let a be a single pixel on a raster map representing the whole study area. S is a set of p points (DHPs) on anisotropic region T with a ∈ T .…”

“…The distance function on T is obtained by taking, for points p and a, the cumulative cost of transition (d a ) from a to p. The two coefficients z and k determine the balance between the size and distance of the DHP. The importance of distance increases in a linear manner while the importance of size increases exponentially (Aichholzer et al 1999). Thus, a larger DHP will be competitively stronger in relation to smaller ones, even at an increased distance.…”

Boat-sail Voronoi diagram and its computation based on a cone-approximation scheme

“…However, the flow of water also displaces the boat by At f (x, y), and hence the actual movement Du of the boat in time interval Zt is represented by .Au = atFvF + At f (x, y). (1) Consequently, the effective speed of the boat in the water flow is given by…”

“…There are still many other distances. They include a geodesic distance [2,11,14], a collision-avoidance distance [3], a distance in a river [20], a skew distance [1], a peeper's distance [6], a crystal-growth distance [12,16], Karlsruhe distance [11], and the ski distance [11], and the Hausdorff distance [6].…”

Time Convex Hull with a Highway

“…Since then, there have been quite a number of research results on variations of this classical problem or related problems, e.g., the Voronoi diagrams [1,2,3,4,5,6,9,10,12,11]. A variation of the convex hull problem is defined as follows.…”