What are the principles, fields, ideas, tools, and techniques that are in play in the development of spatial data structure? There are several:

Geometry. A branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids. With plane geometry we can define a set of polygonal areas with line segments. We can overlay one polygonal set with another, using geometry to calculate where line segments intersect and make new polygons.

Topology. Loosely, a branch of mathematics concerned with the properties of geometric configurations that are unaltered when positions of points, lines, and surfaces are altered. (Classic joke: A topologist is a mathematician who can’t tell the difference between a coffee mug and a doughnut [since each is a solid objects with a single hole].)

Look at the three plane figures composed of lines connected to nodes (see Figure 4-1). Nodes are shown by heavy dots. While configuration “A” and configuration “B” appear to have a lot in common cosmetically, configuration “A” and “C” are topologically identical and Configuration “B” is different from both. “A” and “C” have the same number of lines and nodes as each other, and you can find equivalences in the connections of the nodes in those two configurations. However, you cannot “map” “B” on to either “A” or “C”. If you don’t see this, assign letters to the nodes and numbers to the lines in all three. Make a table for “A,” ...

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