REGION COLORING, EDGE COLORING, AND SCAN-CONVERSION OF MAPS

Abstract

A theory is introduced relating extrinsic colorings of complementary regions of an embedded graph to certain intrinsic colorings of the edges of the graph, called color cycles, that satisfy a certain self-consistency condition. A region coloring is lifted to an edge coloring satisfying the cycle condition, and a dual construction builds a region coloring from any color cycle and any embedding of the graph. Both constructs are canonical, and the constructions are information-conservative in the sense that lifting an arbitrary region coloring to a color cycle and then reconstructing a region coloring from the cycle, using the same embedding, results in the original region coloring. The theory is motivated by, and provides the proof of correctness of, new scan-conversion algorithms that are useful in settings where region boundaries have very high complexity. These algorithms have been implemented and provide useful display functionality previously unavailable on certain rastor devices.