DE-9IM
Encyclopedia
In geometry
, a dimensionally extended nine-intersection model (DE-9IM) is a topological model
used to describe the spatial relation
of two geometries in two-dimensions
(R2). The representation was developed by Clementini and others and is used as a basis for spatial predicate functions
in geographic information systems (GIS) and spatial database
s.
The DE-9IM model is based on a 3×3 intersection matrix
with the form:
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a dimensionally extended nine-intersection model (DE-9IM) is a topological model
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
used to describe the spatial relation
Spatial relation
A spatial relation specifies how some object is located in space in relation to some reference object. Since the reference object is usually much bigger than the object to locate, the latter is often represented by a point. The reference object is often represented by a bounding box .It might be...
of two geometries in two-dimensions
2D geometric model
A 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane.Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and machine parts made of...
(R2). The representation was developed by Clementini and others and is used as a basis for spatial predicate functions
Subroutine
In computer science, a subroutine is a portion of code within a larger program that performs a specific task and is relatively independent of the remaining code....
in geographic information systems (GIS) and spatial database
Spatial Database
A spatial database is a database that is optimized to store and query data that is related to objects in space, including points, lines and polygons. While typical databases can understand various numeric and character types of data, additional functionality needs to be added for databases to...
s.
The DE-9IM model is based on a 3×3 intersection matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
with the form:
-
where dim is the maximum number of dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s of the intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
(∩) of the interiorInterior (topology)In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
(I), boundaryBoundary (topology)In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
(B), and exteriorExterior (topology)In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S. It is itself an open set and is disjoint from S...
(E) of geometries a and b. Empty sets (∅) are denoted as −1 or F (false). Non-empty sets (¬∅) or T (true) are denoted with the maximum number of dimensions of the intersection, specifically 0 for pointsPoint (geometry)In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
, 1 for linesLine (geometry)The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
, 2 for areaAreaArea is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
s. The domainData domainIn data management and database analysis, a data domain refers to all the unique values which a data element may contain. The rule for determining the domain boundary may be as simple as a data type with an enumerated list of values....
of the model also uses * as a wildcard for any condition for dim. The matrix can be represented in a single-line string pattern.
The spatial predicate functions that can be derived from DE-9IM include:Name (synonym) Intersection matrix and code string Equals 
T*F**FFF*Disjoint 
FF*FF****Intersects 
or 
or 
or 
T*********T**********T*********T****Touches (meets) 
or 
or 
FT*******F**T*****F***T****Crosses 
or 
or 
T*T******T*****T**0********Within (inside) 
T*F**F***Contains 
T*****FF*Overlaps 
or 
T*T***T**1*T***T**Covers 
or 
or 
or 
T*****FF**T****FF****T**FF*****T*FF*CoveredBy 
or 
or 
or 
T*F**F****TF**F*****FT*F*****F*TF***
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