Kinetic Voronoi/Delaunay Drawing Tools

We describe two reversible line-drawing methods for cartographic applications based on the kinetic (moving-point) Voronoi diagram. Our objectives were to optimize the user's ability to draw and edit the map, rather than to produce the most efficient batch- oriented algorithm for large data sets, and all our algorithms are based on local operations (except for basic point location). Because the deletion of individual points or line segments is a necessary part of the manual editing process, incremental insertion and deletion is used. The original concept used here is that, as a curve (line) is the locus of a moving point, then segments are drawn by maintaining the topology of a single moving point (MP, or the "pen") as it moves through the topological network (visualized as either the Voronoi diagram or Delaunay triangulation). The trailing line accumulates the adjacency relationships of MP. There are thus three parts to our method: the maintenance of MP in the DT/VD; the use of MP to draw the constrained edges in the Delaunay triangulation; and the use of MP to draw the line segment Voronoi diagram. In all cases deletion is the inverse of the original drawing: move MP so as to "roll up" the desired segment. This approach also has the interesting property that a "log file" of all operations may be preserved, allowing reversion to previous map states, or "dates", as required.