A comparison of neighbourhood relations based on ordinary Delaunay diagrams and area Delaunay diagrams: an application to define the neighbourhood relations of buildings

The aim of this article is to describe a convenient but robust method for defining neighbourhood relations among buildings based on ordinary Delaunay diagrams (ODDs) and area Delaunay diagrams (ADDs). ODDs and ADDs are defined as a set of edges connecting the generators of adjacent ordinary Voronoi cells (points representing centroids of building polygons) and a set of edges connecting two centroids of building polygons, which are the generators of adjacent area Voronoi cells, respectively. Although ADDs are more robust than ODDs, computation time of ODDs is shorter than that of ADDs (the order of their computation time complexity is O(nlogn)). If ODDs can approximate ADDs with a certain degree of accuracy, the former can be used as an alternative. Therefore, we computed the ratio of the number of ADD edges to that of ODD edges overlapping ADDs at building and regional scales. The results indicate that: (1) for approximately 60% of all buildings, ODDs can exactly overlap ADDs with extra ODD edges; (2) at a regional scale, ODDs can overlap approximately 90% of ADDs with 10% extra ODD edges; and (3) focusing on judging errors, although ADDs are more accurate than ODDs, the difference is only approximately 1%.