## 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").

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