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The Dimensionally Extended nine-Intersection Model (DE-9IM) is a topological model and a standard used to describe the spatial relations of two regions (two geometries in two-dimensions, R2), in Geometry, Point-set topology, Geospatial topology, and fields related to computer spatial analysis.

The representation was developed by Clementini and others,[1][2] based on the seminal works of Egenhofer and others[3][4] and was used as a basis for standards of queries and assertions in geographic information systems (GIS) and spatial databases.

The matrix offer an approach to classify geometry relations: roughly speaking, with a true/false matrix domain, there are 512 possible 2D topologic relations, that can be grouped into binary classification schemes. For English speakers, there are about 10 different schemes that have a name, that is, 10 relation types with usual semantics. "Intersects", "Touches", "Equals", and others. When testing two geometries about a scheme, the result of this test is a spatial predicate named by the scheme.

Matrix model[edit]

The DE-9IM model is based on a 3×3 intersection matrix with the form:

\operatorname{DE9IM}(a,b) = \begin{bmatrix}
\dim(I(a) \cap I(b)) & \dim(I(a) \cap B(b)) & \dim(I(a) \cap E(b)) \\
\dim(B(a) \cap I(b)) & \dim(B(a) \cap B(b)) & \dim(B(a) \cap E(b))\\
\dim(E(a) \cap I(b)) & \dim(E(a) \cap B(b)) & \dim(E(a) \cap E(b))

where dim is the maximum number of dimensions of the intersection (∩) of the interior (I), boundary (B), and exterior (E) of geometries a and b.

Note that in this article the words interior and boundary are used in the sense used in algebraic topology and manifold theory, not in sense used in general topology: e. g. by the interior of a line segment we mean the line segment without its endpoints and by its boundary, the two endpoints (in the general topology sense, the interior of a line segment in the plane is empty and the line segment is its own boundary).

In the notation of topological space operators, the matrix elements can be expressed also as

I(a)=ao    B(a)=∂a    E(a)=ae

The dimension of empty sets (∅) are denoted as −1 or F (false). The dimension of non-empty sets (¬∅) are denoted with the maximum number of dimensions of the intersection, specifically 0 for points, 1 for lines, 2 for areas. Then, the domain of the model is {0,1,2,F}.

A simplified version of dim(x) values are obtained mapping the values {0,1,2} to T (true), so using the boolean domain {T,F}. The matrix, denoted with operators, can be expressed as

\operatorname{bin}(\operatorname{DE9IM}(a,b)) = \operatorname{9IM}(a,b) = \begin{bmatrix}
a^o  \cap b^o \ne \emptyset   &   a^o \cap \partial{b} \ne \emptyset & a^o \cap b^e \ne \emptyset \\
\partial{a} \cap b^o\ne\emptyset   &    \partial{a} \cap \partial{b}\ne\emptyset  &  \partial{a} \cap b^e\ne\emptyset \\
a^e \cap b^o\ne\emptyset    &    a^e \cap \partial{b}\ne\emptyset   &   a^e  \cap b^e\ne\emptyset

Both matrix forms, with dimensional and boolean domains, can be serialized as "DE-9IM string codes", that is, can be represented in a single-line string pattern. Since 1999 the string codes have a standard[5] format for database analysis.

For output checking or pattern analysis, a matrix value (or a string code) can be checked by a "mask": a desired output value with optional asterisk symbols as wildcards — that is, "*" indicating output positions that the designer does not care about (free values or "don't-care positions"). Then, the mask's domain is {0,1,2,T,F,*}, or {T,F,*} for the boolean form.

The simplified models, 4-Intersection and 9-Intersection, proposed before DE-9IM for express spatial relations[6] (and originating the labels 4IM and 9IM) can replace the DE-9IM to optimize calculations, when input conditions satisfy specific constraints.


Visually, for two overlapping polygonal geometries, this looks like:[7]

b   De9im04.png
Interior Boundary Exterior
Interior De9im05.png

  dim[I(a)I(b)] = 2  


  dim[I(a)B(b)] = 1  


  dim[I(a)E(b)] = 2  


  dim[B(a)I(b)] = 1  


  dim[B(a)B(b)] = 0  


  dim[B(a)E(b)] = 1  


  dim[E(a)I(b)] = 2  


  dim[E(a)B(b)] = 1  


  dim[E(a)E(b)] = 2  

Reading from left-to-right and top-to-bottom, the resulted DE-9IM(a,b) string code is '212101212', a compacted form for say "II=2, IB=1, IE=2, BI=1, BB=0, BE=1, EI=2, EB=1, EE=2".

Spatial predicates[edit]

The model express important space relations because they are invariant to rotation, translation and scaling transformations; but, in its most general form the DE-9IM model is too complex to make it usable. So, the adoption of "named predicates" have been specified.

Spatial predicates are binary invariant space relations, with more usual semantics. The spatial predicate functions that can be derived (expressed by masks) from DE-9IM include:[4] [8]

II & IB & IE \\
BI & BB & BE \\
EI & EB & EE

Predicates defined with masks of domain {T,F,*}

Name (synonym) Intersection matrix and mask code string
(boolean OR between matrices)
Meaning and definition,[4][9] Same
Equals \Bigl[\begin{smallmatrix}
II ∧ ~IE ∧ ~BE ∧ ~EI ∧ ~EB
a and b are topologically equal. "Two geometries are topologically equal if their interiors intersect and no part of the interior or boundary of one geometry intersects the exterior of the other".[10]
Within & Contains
Disjoint \Bigl[\begin{smallmatrix}
~II ∧ ~IB ∧ ~BI ∧ ~BB
a and b are disjoint, have no point in common. They form a set of disconnected geometries.
not Intersects
Touches (meets) \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
(~IIIB) ∨ (~IIBI) ∨ (~IIBB)
a touches b, they have at least one boundary point in common, but no interior points.
FT******* F**T***** F***T****
Contains \Bigl[\begin{smallmatrix}
II ∧ ~EI ∧ ~EB
b is within a. Geometry b lies in the interior of a. Another definition: "a 'contains' b iff no points of b lie in the exterior of a, and at least one point of the interior of b lies in the interior of a".[11]
Covers \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
(II ∧ ~EI ∧ ~EB) ∨ (IB ∧ ~EI ∧ ~EB) ∨ (BI ∧ ~EI ∧ ~EB) ∨ (BB ∧ ~EI ∧ ~EB)
a covers b. Geometry b lies in the interior of a (extends Contains). Other definitions: "no points of b lie in the exterior of a", or "Every point of b is a point of (the interior of) a".
T*****FF* *T****FF* ***T**FF* ****T*FF*

Predicates that can be obtained by the above, by logic negation or parameter inversion (matrix transposition):

Intersects \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] a intersects b: geometries a and b have at least one point in common. not Disjoint
T******** *T******* ***T***** ****T****
\end{smallmatrix}\Bigr] a is within b, a lies in the interior of b. Contains(b,a)
CoveredBy \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] a is covered by b (extends Within): every point of a is a point of b, and the interiors of the two geometries have at least one point in common. Covers(b,a)
T*F**F*** *TF**F*** **FT*F*** **F*TF***

Predicates that checks the input dimensions, and are defined with masks of domain {0,1,T,F,*}

dim(a)≠dim(b) or
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] a crosses b, they have some but not all interior points in common (and the dimension of the intersection is less than that of at least one of them). Mask selection rules are checked only when dim(a)≠dim(b), except by point/point or line/line inputs, otherwise is false:[12]
(II=0) for points or lines,   (IIIE) when dim(a)<dim(b),   (IIEI) when dim(a)>dim(b)
\end{smallmatrix}\Bigr] \Bigl[\begin{smallmatrix}
\end{smallmatrix}\Bigr] a overlaps b, they have some but not all points in common, they have the same dimension, and the intersection of the interiors of the two geometries has the same dimension as the geometries themselves. Mask selection rules are checked only when dim(a)=dim(b), otherwise is false:
(IIIEEI) for points or surfaces,   (II=1 ∧ IEEI) for lines
dim=0 or 2

Notice that:

  • The topologically equal definition does not imply that they have the same points or even that they are of the same class.
  • The output of DE-9IM(a,b) have the information contained in a list of all interpretable predicates about geometries a and b.
  • All predicates are computed by masks, only Crosses and Overlaps has adicional conditions about dim(a) and dim(b).
  • All mask string codes ends with '*'. It is because EE have no extra information.
  • The Equals mask, T*F**FFF*, is the "merge" of Contains (T*****FF*) and Within (T*F**F***): (II ∧ ~EI ∧ ~EB) ∧ (II ∧ ~IE ∧ ~BE).
  • There are no mask for situations involving complex types, like a Point/Multipoint situation. Example: with the above definition the code 0FFFFF0F2 have the Crosses predicate (satisfies the mask T*****T**), but by a more rigorous definition, like the JTS definition, not.[13]
  • The mask T*****FF* is within the definition of both, Contains and Covers. Covers is a more inclusive relation. In particular, unlike Contains it does not distinguish between points in the boundary and in the interior of geometries. For most situations, Covers should be used in preference to Contains.[9]
  • Similarly, the mask T*F**F*** falls within the definition of both, Within and CoveredBy.


Examples of spatial relations.

The terminology used for translating the nine relations into more usual semantics, is based on reasonable conventions and the tradition of topological studies.[4] Relationships (between two geometries a and b) such as Intersects, Disjoint, Touches, Within, Equals, have an obvious semantic:[11][14]

  • Equals:   a = b    that is    (ab = a) ∧ (ab = b)
  • Within:   ab = a
  • Intersects:   ab ≠ ∅
  • Touches:   (ab ≠ ∅) ∧ (aοbο = ∅)

The other ones, Covers, Contains, CoveredBy and Within, have subtle aspects to their definition which are contrary to intuition.

Example of "non-obvious predicates", that has an aspect of its definition which may produce unexpected behaviour:[11] a line L which is completely contained in the boundary of a polygon P is not considered to be contained in P. This quirk can be expressed as "Polygons do not contain their boundary". See the Contains definition above: the last clause, "at least one point of the interior of B lies in the interior of A", causes the trap. In this case, the predicate Covers has the intuitively expected semantics (see definition), avoiding boundary considerations.

For a better intuitive understand we can use the dimensionality of inputs, as justification to a gradual introduction of semantic complexity:

Relations between Appropriate predicates Semantic added
point/point Equals, Disjoint Other valid predicates collapses into Equals.
point/line adds Intersects Intersects is a flexibilization of Equals, "some equal point at the line".
line/line adds Touches, Crosses, ... Touches is a constraint of Intersects, about "only boundaries"; Crosses about "only one point".

Coverage on possible matrix results[edit]

The number of possible results in a boolean 9IM matrix is 29=512, and in a DE-9IM matrix is 39=6561. The probability of one of these results come to satisfy a specific predicate is determined as following,

  • 93.7% Intersects;
  • 43.8% Touches;
  • 25% Crosses (for valid inputs, 0% otherwise);
  • 23.4% Covers and CoveredBy;
  • 12.5% Contains, Overlaps (for valid inputs, 0% otherwise) and Within;
  • 6.3% Disjoint;
  • 3.1% Equals.

On usual applications the geometries intersects a priori, and the another relations are checked.

The composite predicates "Intersects OR Disjoint" and "Equals OR Different" have the sum 100% (always true predicates), but "Covers OR CoveredBy" have 41%, that is not the sum, because they are not logical complements neither independent relations; idem "Contains OR Within", that have 21%. The sum 25%+12.5%=37.5% is obtained when ignoring overlaping of lines in "Crosses OR Overlaps", because the valid input sets are disjoints.

Queries and assertions[edit]

The DE-9IM offers a full descriptive assertion about the two input geometries. It is a mathematical function that represents a complete set of all possible relations about two entities, like a Truth table, the Three-way comparison, a Karnaugh map or a Venn diagram. Each output value is like a truth table line, that represent relations of specific inputs.

As illustrated above, the output '212101212' resulted from DE-9IM(a,b) is a complete description of all topologic relations between specific geometries a and b. It say to us that "II=2, IB=1, IE=2, BI=1, BB=0, BE=1, EI=2, EB=1, EE=2".

By other hand, if we check predicates like Intersects(a,b) or Touches(a,b) — for the same example we have "Intersects=true and Touches=true" — it is an incomplete description of "all topologic relations". Predicates also do not say any thing about the dimensionality of the geometries (it doesn't matter if a and b are lines, areas or points).

This independence of geometry-type and the lack of completeness, on predicates, are useful for general queries about two geometries:

(interior/boundary/exterior semantic) (usual semantic)
Assertions: more descriptive
" a and b have DE-9IM(a,b)='212101212' "
less descriptive
" a Touches b "
Queries: more restrictive
" Show all pair of geometries where DE-9IM(a,b)='212101212' "
    more general
" Show all pair of geometries where Touches(a,b) "

For usual applications, the use of spatial predicates also is justified by being more human-readable than DE-9IM descriptions: a typical user have better intuition about predicates (than a set of interiors/border/exterior intersections).

Predicates have useful semantic into usual applications, so it is useful the translation of a DE-9IM description into a list of all associated predicates,[15][16] that is like a casting process between the two different semantic types. Examples:

  • The string codes "0F1F00102" and "0F1FF0102" have the semantic of "Intersects & Crosses & Overlaps".
  • The string code "1FFF0FFF2" have the semantic of "Equals".
  • The string codes "F01FF0102", "FF10F0102", "FF1F00102", "F01FFF102", and "FF1F0F1F2" have the semantic of "Intersects & Touches".


The Open Geospatial Consortium (OGC) has standardized the typical spatial predicates (Contains, Crosses, Intersects, Touches, etc.) as boolean functions, and the DE-9IM model,[17] as a function that returns a string (the DE-9IM code), with domain of {0,1,2,F}, meaning 0=point, 1=line, 2=area, and F="empty set". This DE-9IM string code is a standardized format for data interchange.

The Simple feature access (ISO 19125) standard,[18] in the chapter 7.2.8, "SQL routines on type Geometry", recommends as supported routines the SQL/MM Spatial[19] (ISO 13249-3 Part 3: Spatial) ST_Dimension, ST_GeometryType, ST_IsEmpty, ST_IsSimple, ST_Boundary for all Geometry Types. The same standard, consistent with the definitions of relations in "Part 1, Clause" of the SQL/MM, recommends (shall be supported) the function labels: ST_Equals, ST_Disjoint, ST_Intersects, ST_Touches, ST_Crosses, ST_Within, ST_Contains, ST_Overlaps and ST_Relate.

The DE-9IM in the OGC standards use the following definitions of Interior and Boundary, for the main OGC standard geometry types:[20]

Subtypes Dim Interior (I) boundary (B)
Point, MultiPoint 0 Point, Points Empty
LineString, Line 1 Points that are left when the boundary points are removed. Two end points.
LinearRing 1 All points along the geometry. Empty.
MultilineString 1 Points that are left when the boundary points are removed. Those points that are in the boundaries of an odd number of its elements (curves).
Polygon 2 Points within the rings. Set of rings.
MultiPolygon 2 Points within the rings. Set of rings of its elements (polygons).
NOTICE: exterior points (E) are points p not in the interior or boundary, so not need extra interpretation, E(p)=not(I(p) or B(p)).

Implementation and practical use[edit]

Most spatial databases, such as PostGIS, implements the DE-9IM() model by the standard functions:[21] ST_Relate, ST_Equals, ST_Intersects, etc. The function ST_Relate(a,b) outputs the standard OGC's DE-9IM string code.

Examples: two geometries, a and b, that intersects and touches with a point (for instance with dim(B(a)∩I(b))=0 and dim(I(a)∩I(b))=F), can be st_relate(a,b)='FF1F0F1F2' or st_relate(a,b)='FF10F0102' or st_relate(a,b)='FF1F0F1F2'. It also satisfies st_intersects(a,b)=true and st_touches(a,b)=true. When ST_Relate(a,b)='0FFFFF212', the returned DE-9IM code have the semantic of "Intersects(a,b) & Crosses(a,b) & Within(a,b) & CoveredBy(a,b)", that is, returns true on the boolean expression st_intersects(a,b) AND st_crosses(a,b) AND st_within(a,b) AND st_coveredby(a,b).

The use of st_relate() is faster than direct computing of a set of correspondent predicates.[22] There are cases where the use of st_relate() is the unique access form of a complex predicate — see the example of the code 0FFFFF0F2,[13] of a point that not "crosses" a multipoint (a object that is a set of points), but predicate Crosses (when defined by a mask) returns true.

It is usual also to overload the st_relate() by a mask parameter, or use a returned st_relate(a,b) string into the st_relateMatch() function.[23] When using st_relate(a,b,mask), it returns a boolean. Examples:

  • ST_Relate(a,b,'*FF*FF212') returns true when ST_Relate(a,b) is 0FFFFF212 or 01FFFF212, and returns false when 01FFFF122 or 0FF1FFFFF.
  • ST_relateMatch('0FFFFF212','*FF*FF212') and ST_relateMatch('01FFFF212','TTF*FF212') are true, ST_relateMatch('01FFFF122','*FF*FF212') is false.


  • "Egenhofer-Matrix" is a synonym for the 9IM 3x3 matrix of boolean domain.[24]
  • "Clementini-Matrix" is a synonym for the DE-9IM 3x3 matrix of {0,1,2,F} domain.[24]
  • "Egenhofer operators" and "Clementini operators" are sometimes a reference to matrix elements as II, IE, etc. that can be used in boolean operations. Example: the predicate "G1 contains G2" can be expressed by "<G1| II ∧ ~EI ∧ ~EB |G1>", that can be translated to mask syntax, "T*****FF*".
  • Predicates "meets" is a synonym for touches; "inside" is a synonym for within; Oracle's[16] "ANYINTERACT" is a synonym for intersects, "OVERLAPBDYINTERSECT" is a synonym for overlaps, and "OVERLAPBDYDISJOINT" have not a synonym.

See also[edit]

Standards:       Software:       Related topics


  1. ^ Clementini, Eliseo; Di Felice, Paolino; van Oosterom, Peter (1993). "A small set of formal topological relationships suitable for end-user interaction". In Abel, David; Ooi, Beng Chin. Advances in Spatial Databases: Third International Symposium, SSD '93 Singapore, June 23–25, 1993 Proceedings. Lecture Notes in Computer Science. 692/1993. Springer. pp. 277–295. doi:10.1007/3-540-56869-7_16. 
  2. ^ Clementini, Eliseo; Sharma, Jayant; Egenhofer, Max J. (1994). "Modelling topological spatial relations: Strategies for query processing". Computers & Graphics 18 (6): 815–822. doi:10.1016/0097-8493(94)90007-8. 
  3. ^ Egenhofer, M.J.; Franzosa, R.D. (1991). "Point-set topological spatial relations". Int. J. GIS 5 (2): 161–174. doi:10.1080/02693799108927841. 
  4. ^ a b c d Egenhofer, M.J.; Herring, J.R. (1990). A Mathematical Framework for the Definition of Topological Relationships. 
  5. ^ The "OpenGIS Simple Features Specification For SQL", Revision 1.1, was released at May 5, 1999. It was the first international standard to establish the format conventions for DE-9IM string codes, and the names of the "Named Spatial Relationship predicates based on the DE-9IM" (see section with this title).
  6. ^ M. J. Egenhofer, J. Sharma, and D. Mark (1993) "A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis", In: Auto-Carto XI.
  7. ^ Chapter 4. Using PostGIS: Data Management and Queries
  8. ^ JTS: Class IntersectionMatrix, Vivid Solutions, Inc. 
  9. ^ a b Geometry
  10. ^ JTS Technical Specifications of 2003.
  11. ^ a b c M. Davis (2007), "Quirks of the 'Contains' Spatial Predicate".
  12. ^ ST_Crosses
  13. ^ a b JTS test case of "point A within one of B points",
  14. ^ Câmara, G.; Freitas, U. M. (1995). Fields and Objects Algebras for GIS Operations. CiteSeerX: 
  15. ^ A DE-9IM translator, of all associated predicates of a spatial relation.
  16. ^ a b Note. The Oracle's spatial funcion SDO_RELATE() do only a partial translation, internally, offering to user a mask for a or-list of predicates to be checked, instead the DE-9IM string.
  17. ^ "OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 2: SQL option", OGC,
  18. ^ Open Geospatial Consortium Inc. (2007), "OpenGIS® Implementation Standard for Geographic information - Simple feature access - Part 2: SQL option", OGC document 06-104r4 version 1.2.1 (review of 2010-08-04).
  19. ^ ISO 13249-3 Part 3: Spatial, summarized in SQL Multimedia and Application Packages (SQL/MM).
  20. ^ "Encyclopedia of GIS", edited by Shashi Shekhar and Hui Xiong. SpringerScience 2008. pg. 242
  21. ^ ST_Relate() PostGIS function online documentation.
  22. ^ Chapter 4. Using PostGIS: Data Management and Queries
  23. ^ ST_RelateMatch() PostGIS function online documentation.
  24. ^ a b "Encyclopedia of GIS", S. Shekhar, H. Xiong. ISBN 978-0387359755.

External links[edit]