- The DRAFT - NATO ARW - AUGUST 1989 Spatial Data Structures Extension from One to Two Dimensions

The field of Computing Science has, for some years, developed considerable expertise in solving problems based on onedimensional linked lists: these form the basis for many searching, sorting, graph and tree procedures. Graphs, based on individual edge-links, have been used extensively to describe two-dimensional polygon information, as a collection of nodes, arcs and regions. Once created, these are readily traversed by standard methods. Creation of these and similar structures from co-ordinate information is, however, still a problem. Comparison of the differences between raster and vector data suggests that one major advantage of the raster mode is its spacecovering property, permitting direct reference to adjoining cells. This property is possible for any tiling of the plane, where polygonal regions abut each other, and boundaries meet in (typically) triple junctions. These polygonal adjacencies may be expressed in the dual triangulation. For arbitrarily-distributed objects in the plane initially no such adjacency relationships exist, but they can be created by the generation of the Voronoi diagram. This has been readily achievable for points, and the extension to line segments permits the representation of arbitrarily-complex objects in a map. The dual triangulation of the Voronoi polygons forms the Voronoi adjacency graph (VAG) which expresses the "natural" adjacency relationships of unconnected objects. Any polygon set in the plane may be modified locally by splitting or merging of existing polygons, and this is equivalent to the insertion or deletion of triangle pairs in the dual. These triangles express the adjacencies of map objects, and can be traversed as graphs to perform various searches, etc. Each triangle has pointers to three adjacent triangles, and thus forms two-dimensional triply-linked lists that may be manipulated by conventional computing techniques. Direct parallels exist between operations on 2-D triply-linked lists an 1-D doubly-linked lists used for searching, sorting, insertion and deletion of collections of objects. It is anticipated that equivalent algorithms can be developed for a variety of spatial processes. BACKGROUND AND MOTIVATION Can Vectors Be Rehabilitated? I would like to confess to a strong personal bias in recent years I have felt that vector methods in spatial modelling have had a raw deal, and were frequently being taken over by raster techniques. However, the more I looked at raster methods, the less satisfied I, was with what they were trying to achieve. The superficial attractions were obvious, but underneath, let's face it, rasters were degenerate (in what sense I will describe later). Some of the reasons for my prejudices came from my early interest in interpolation problems, where some kind of function is described about a portion of the map, and elevations estimated at grid nodes for subsequent contouring. The more I looked at this problem the less I was satisfied with almost ANY of the underlying assumptions, and the arbitrary techniques and approximations used. To name merely a few: there is no need to estimate intermediate values onto a grid in fact it is counter-productive: polynomial functions in Cartesian coordinates create massive problems in smooth surface continuity; the selection of a set of neighbouring points for local approximation or averaging can never be achieved successfully (i.e. without surface discontinuities due to arbitrary acceptance or rejection of points) using an exclusively metric criterion; and any weighting function used to average these neighbours will not be successful if exclusively metric in nature. There had to be a better, a more "natural" approach. A similar feeling occurred with the examination of G.I.S. (Geographic Information Systems) problems, where spatial neighbourhood/adjacency issues were combined with data base concepts. It is clearly desirable that one object in the data base be related to one object in the spatial referencing system but in a raster system space is sliced into little squares which then have to be allocated to objects (with limited resolution) or else in vector systems a great deal of overhead is expended on connecting little pieces of lines that (approximately) meet each other, with considerable difficulties in extracting the graph network (topology) from the coordinate information (geometry). The old familiar feeling recurred when working with finite-element or finite-difference models of groundwater flow, and similar problems. Either you live with a very coarse grid-based representation of your aquifer or rock formation, or you create a system of equations and spatial elements that even hardened practitioners agree is not intuitively simple. In the last few years, therefor, I have been trying to visualize my "ideal" system for spatial data manipulation, and I have been looking for simple expressions of spatial relationships that even a pigeon (or a computer) could understand. Some thoughts on this subject follow. In two dimensions, the only things that could easily be agreed to be next to each other would be pairs of regions (polygons) with a common boundary. All other cases would be subject to some level of disagreement. Secondly, in two dimensions and ignoring the insane pre-occupation of homo sapiens with right-angles, all regions meet at triple junctions. Another observation is that in real life space is continuous if the pencil is removed from the table the space it occupied is still there and not as a residue of little squares, either. In other words, polygons are space-filling, and hence adjacency is a meaningful concept. Vectors (thought of as collections of disconnected matchsticks) are not space-filling and hence have great difficulty in communicating with their neighbours (if they have any). They form, however, real objects on the map road segments, etc., or polygon boundaries. Following this line of thought, it could be considered desirable to create a space-covering tiling (or polygon set) about the individual map objects. (For the purposes of the following discussion it will be assumed that map objects are described by, and made up of, points and line-segments, and these primitive elements, or atoms, are therefore our primary concern.) If each point or line-segment has a single polygon or tile associated with it, producing a space-covering set, then by this method spatial . adjacency between originally disconnected objects could be unambiguously established. In addition, the boundary between two adjacent polygons is by definition the expression of an adjacency relationshin between those two polygons and therefore the two objects forming the "nuclei" of these polygons have the specified boundary relationship between them. Thus, since polygons can be considered to meet at triple junctions, between all map objects forms the dual trianqulation of the Given this concept of a continuous polygon coverage expressing the adjacency relationships between map objects, it merely remains to give the polygon definition rules for any given set of map objects. The most obvious (but not the only) definition is the . Voronoi tessellation of the set of points plus line segments forming the map, probably assuming the nearest-neighbour, Euclidean metric Voronoi diagram. The main distinction between the Voronoi tessellation and the general polygon case is that the Voronoi boundaries may be directly re-constituted from the original map objects and the relationship triangulation, thus permitting the conversion of either representation to the corresponding dual form as desired for any particular map operation. It has proven convenient to preserve all spatial relationships in the triangulation form, as these are fixed-length records, and to perform all searches and data-set modifications in the polygon form. While these are equivalent operations in many cases, the visualization is frequently simpler in one mode or the other. This then is a progress report on attempts to achieve a variety of operations on spatial information (points, lines and areas) within the constraints of certain rules: 1) No "graph paper" is to be used conventional Cartesian coordinate systems are to be avoided: there will be no hidden grids, slopes or axes. Coordinates will only be used for calculation of distances (i.e. a "metric"). 2) Space is to be considered as continuous: at all times any particular location will be assigned to some defined polygon or element of a space-covering tiling. 3) The only acceptable definition of adjacency is that of two polygons having a common boundary.' 4) Operations on this data structure should be discrete equivalent to one-dimensional linked list operations. As will be shown, the two tools that form the basis for this approach, that extend the problem from a general arbitrary tiling or polygon set to any set of objects on a map, are the use of Voronoi regions and dual (triangulation) representation of spatial