WEEK 3:
- The Jordan Curve Theorem:
- permits correct assignment of addresses on either sides of streets
- permits visually appropriate coloring of polygons
- illustrates the need to split complex curves apart at nodes where the curve crosses itself in order to ensure that the two properties above will hold on maps. This fact is important in digitizing (and elsewhere).
- The Four Color Theorem:
- shows that four colors are enough to color any map in the plane with adjacency of regions defined across an edge only.
- suggests that if distinctions need to be made that correspond to gradations in data, that a graded scale of four, and often more, colors will make appropriate distinctions among regions
- The One-point Compactification Theorem:
- shows that the skin of a spherical globe cannot be perfectly flattened into the plane; it fails to do so by at least one point.
- that there (therefore) can be no perfect map in the plane.
Useful extension: go to www.usgs.gov and search the site for "spatial tools" and for "animal movement." These tools are free and are run along with Spatial Analyst Extension to ArcView. |