Topology

Topology:

In 1736, the mathematician Leonhard Euler published a paper that arguably started the branch of mathematics known as topology. More recently, the United States Census Bureau, while preparing for the 1970 census, pioneered the application of mathematical topology to maps to reduce the errors in tabulating massive amounts of census data. Today, topology in GIS is generally defined as the spatial relationships between adjacent or neighboring features.

Mathematical topology assumes that geographic features occur on a two-dimensional plane. Through planar enforcement, spatial features can be represented through nodes (0-dimensional cells); edges, sometimes called arcs (one-dimensional cells); or polygons (two-dimensional cells). 

In GIS, topology is implemented through data structure. An ArcInfo coverage is a familiar topological data structure. A coverage explicitly stores topological relationships among neighboring polygons in the Arc Attribute Table (AAT) by storing the adjacent polygon IDs in the LPoly and RPoly fields. Adjacent lines are connected through nodes, and this information is stored in the arc-node table. The ArcInfo commands, CLEAN and BUILD, enforce planar topology on data and update topology tables.

Over the past two or three decades, the general consensus in the GIS community had been that topological data structures are advantageous because they provide an automated way to handle digitizing and editing errors and artifacts; reduce data storage for polygons because boundaries between adjacent polygons are stored only once; and enable advanced spatial analyses such as adjacency, connectivity, and containment. Another important consequence of planar enforcement is that a map that has topology contains space-filling, nonoverlapping polygons. Consequently, so-called cartographic (i.e., nontopological) data structures are no longer used by mainstream GIS software.

·         Topology rules help create datasets with greater integrity - (i.e. no slivers between polygons, no unsnapped nodes between lines that should be connected, no twisted line features).

·         Topology facilitates the editing of shared features between different spatial layers.

·         Different data format have different implementations of topology with varying degrees of functionality. With data formats supported by ArcMap, Geodatabses have the greatest topological functionality.

·         Not every GIS project really requires topology. If you're just making a map of city locations and roads, then you don't really need topology. If you want to find the optimal path between five different cities, then topology is useful, but there are plenty of GIS projects where you don't really need topology (or at least topology built into the datasets).

Generally, topology is employed to do the following:

·         Manage coincident geometry (constrain how features share geometry). For example, adjacent polygons, such as parcels, have shared edges; street centerlines and the boundaries of census blocks have coincident geometry; adjacent soil polygons share edges; etc.

·         Define and enforce data integrity rules (such as no gaps should exist between parcel features, parcels should not overlap, road centerlines should connect at their endpoints).

·         Support topological relationship queries and navigation (for example, to provide the ability to identify adjacent and connected features, find the shared edges, and navigate along a series of connected edges).

·         Support sophisticated editing tools that enforce the topological constraints of the data model (such as the ability to edit a shared edge and update all the features that share the common edge).

·         Construct features from unstructured geometry (e.g., the ability to construct polygons from lines sometimes referred to as "spaghetti").