A NEW MAP TRANSFORMATION METHOD FOR HIGHLY DEFORMED MAPS BY CREATING HOMEOMORPHIC TRIANGULATED IRREGULAR NETWORK
Inoue1, M. Wako2,
There are several needs for map transformation which is applicable to highly deformed maps.
Typical examples of highly deformed maps are historical maps which position accuracy is low. The transformation of historical map to correct geometric deformation enables to overlay them on present maps and to compare the change of land use, allocation of streets, and so on.
Other examples are cartograms. If it is possible to transform the spatial distribution of geographic features on cartograms, it would enhance the applicability of visualization by cartograms. One type of cartograms is area cartogram, which represents data by the size of area on the map. If spatial distribution of hospitals is transformed on population area cartograms, in other words equal population density maps, it could visualize the regional disparity of medical services apparently. Another type of cartograms is distance cartogram, which displays the proximity of points by the distances on the map. Usually distance cartograms are described by points and lines and sometimes not easily recognizable. The transformation of boundaries of countries on distance cartograms would increase their readability.
The map transformation is useful as described above; however, when the map deformation is large and the number of control points is limited, it is difficult to perform homeomorphic map transformation. Moreover, the topology of point configurations collapses in some cases on distance cartograms, since the proximity data space is different from the geographic space. The transformation of these highly deformed maps is impossible without deleting some control points. In this study, we propose a simple algorism for homeomorphic transformation by deleting some control points.
The input is the coordinates of control points on the reference and target maps. First it performs the Delaunay triangulation using the control points on the reference map, and creates the same network on the target map. If there are any overlapped triangles on the target map, flip the triangles’ edges to reduce overlapped area. If it is impossible to remove overlap, finally it removes the control points that disturb to create the homeomorphic network. Then it applies transformation proposed by Akima (1978).
The numerical experiences show that the proposed algorithm runs quite fast, and the output results show that maps are smoothly fitted to the most of control points.