Geoffrey Dutton's SDH 96 Paper

Planetary geocoding using polyhedral tessellations are a concise and elegant way to organize both local and global geospatial data that respects and documents locational accuracy. After a brief review of several such spatial referencing systems, topological, computational, and geometric properties of one of them are examined. The particular model described in the remainder of the paper – the octahedral quaternary triangular mesh (O-QTM) – is being developed to handle and visualize vector-format geodata in a hierarchical triangulated domain. The second section analyzes the geometric regularity of the model, showing that its facets are relatively similar, having vertices spaced uniformly in latitude and longitude, and areas that vary by less than 42 % from their mean sizes. Section 3 describes some fundamental operations on this structure, including mapping from geographic coordinates into O-QTM addresses and back again, filtering map detail through the triangular hierarchy and associating locations that are close together, but in different branches of the tree structure. The final section outlines and illustrates a recent application of O-QTM to map generalization, using its multi-resolution properties to enable multiple cartographic representations to be built from a single hierarchical geospatial database. 1 Hierarchical Polyhedral Modeling of Planetary Locations Mapmakers and others have attempted to model the earth as a polyhedron for many years, going back to at least the time of the German artist Albrecht Dürer (1471-1528), whose drawings of polyhedral globes appear to be the first instance of thinking about mapping the planet in this way. In the late 19th and early 20th centuries, a number of cartographers, such as Cahill (Fisher and Miller, 1944; also used by Lugo and Clarke, 1995 in their variant of QTM), reinvented this idea, projecting the land masses of the Earth to various polyhedra, then unfolding their facets into flat, interrupted maps. The best known of these is R. B. Fuller's Dymaxion projection, dating from the early 1940's (Unknown, 1943). Originally based on a cubeoctohedron, the Dymaxion Map was then recast as an icosahedron, oriented to the earth in a way that minimized the division of land areas between its 20 facets. Fuller devised a projection method -only recently well-enough understood to implement digital algorithms for it (Gray, 1994; also see Snyder, 1992) -that has remarkably little distortion. Pre-computer polyhedral projections all were based on platonic or other simple shapes, and were intended as amusements or devices that let a paper globe be unfolded to lie flat. They did not attempt to deal with more complicated cases involving subdivision of polyhedral facets into smaller ones that would more closely fit the figure of the Earth and have more inherent accuracy. Fuller's geodesic domes are examples of how a physical polyhedron can be subdivided to give it greater physical strength per unit weight and span larger areas than any simple polyhedron is capable. Fuller apparently never treated the Dymaxion map in the same manner, probably because he viewed it as a physical structure that was nearly optimal for display of global thematic data, and bringing it to the next level of complexity (either 60 or 80 facets, depending on the method of subdivision) would make the map unwieldy to manufacture and manipulate, and achieve no particular benefit. Now, it’s a different world. The digital revolution in mapping and associated geoprocessing techniques have freed mapmakers from such practical constraints and physical limitations, and over the past 30 years a number of approaches have blossomed for using polyhedra to index and display spatial data on a world-wide basis for a variety of purposes. Several such models are summarized in order to indicate how they are alike and in what respects their form and properties differ. 1.1 The Octahedral Quaternary Triangular Mesh (O-QTM) Framework Inspired by Fuller's maps and domes, the author developed a global digital elevation model in 1983 that divided a planet into facets defined by a concentric octahedron and cube. (Dutton, 1984). In 1988 this model was revisited, revised and recast as a tool for spatially indexing planimetric data in a geographic information system (Dutton, 1989). The cube was discarded, but the octahedron remained as a geometric basis that roots a forest of eight quadtrees containing roughly equal triangular quadrants (facets) that approximate a sphere quite closely after only a few subdivisions. Figure 1a depicts the basis of the O-QTM model, an octahedron embedded in a spheroid. Figure 1b illustrates the quadrant numbering scheme, using a map projection that renders every facet at each level as an isosceles right triangle. Use of this projection simplifies the computation of addresses, as figure 3 shows.