Spatial Thinking in Planning Practice: An Introduction to GIS

21 CHAPTER 3: TOPOLOGY AND CREATING DATA GEOMETRIC PRIMITIVES Topology is the sub!eld of mathematics that deals with the relationship between geometric entities, speci!cally with properties of objects that are preserved under continuous deformation. A GIS topology is a set of rules and behaviors that model how points, lines, and polygons share coincident geometry. "e concepts of topology are very useful for geographers, surveyors, transportation specialists, and others interested in how places and loca- tions relate to one another. We have learned that a location is a zero-dimensional entity (it has no length, width, height, or volume), locations alone are not su&cient for representing the complexity of the real world. Locations are frequently composed into one or more geometric primitives, which include the set of entities more common- ly referred to as: 1. Points; 2. Lines; and 3. Polygons (or Areas). In the !eld of Topology, we can expand them to: 1. Nodes: zero-dimensional entities represented by coordinate pairs. Coordinates for nodes may be x,y values like those in Euclidean geometry or longitude and latitude coordinates that represent places on Earth’s sur- face. In both cases, a third z value is sometimes added to specify a location in three dimensions; 2. Edges: one-dimensional entities created by connecting two nodes. "e nodes at either end of an edge are called connecting nodes and can be referred to more speci!cally as a start node or end node, depending on the direction of the edge, which is indicated by arrowheads. Edges in TIGER have direction so that the le$ and right side of the street can be determined for use in address matching. Nodes that are not associated with an edge and exist by themselves are called isolated nodes. Edges can also contain vertices, which are optional intermediate points along an edge that can de!ne the shape of an edge with more speci!city than start and end nodes alone. Examples of edges encoded in TIGER are streets, railroads, pipelines, and rivers; and 3. Faces: two-dimensional (length and width) entities that are bounded by edges. Blocks, counties, and vot- ing districts are examples of faces. Since faces are bounded by edges and edges have direction, faces can be designated as right faces or le$ faces. Figure below shows an example of these geometric primitives in a realistic arrangement. In this example, note that: 1. Nodes N14 and N17 are isolated nodes; 2. N7 and N6 are the start and end nodes of edge E1; and 3. Due to the directionality of edges, face F2 is on the le$ of edge E8.