rNT. J. GEocRApHrcAL TNFoRMATToNsysrEMS, 1991, vor-. 5, No. 2, 161 174
:FRSIS)
ie fuzzy
r is then
lational
Point-set topological spatial relations
MAX J. EGENHOFER
ridesan
natural
:hensive
blemsof
datato
s heldin
benefits
,ationof
He also
lap may
resource
t ofGIS
y'which
es.They
tomated
;ationof
l datahe
a library
nlikethe
richmap
gwledSe
[rredata
ltionsfor
it merits.
fPocK
IPTILL
National Center for Geographic Information and Analysis and
Department of Surveying Engineering, 107 Boardman Hall,
University of Maine, Orono, Maine 04469,U.S.A.
and ROBERT D. FRANZOSA
Department
of Mathematics,
417 NevilleHall,
Universityof Maine,Orono,Maine04469,U.S.A.
Abstract. Practicalneedsin geographic
(GIS)haveledto the
informationsystems
investigation
of formaland soundmethodsof describingspatialrelations.After an
introduction to the basic ideas and notions of topology, a novel theory of
topologicalspatialrelationsbetweensetsis developedin which the relationsare
delinedin termsof the intersections
of the boundariesand interiorsof two sets.By
considering
enrpty andnon-empty
asthevaluesof theintersections,
a total of sixteen
topologicalspatialrelationsis described,
eachof whichcan be realizedin R2.This
setis reducedto nine relationsif the setsare restrictedto spatialregions,a fairly
broadclassofsubsets
ofa connected
topological
spacewith in appliJation
to GIS.
It is shownthat theserelationscorrespondto someof the standardsettheoretical
and topologicalspatialrelationsbetweensetssuch as equality,disjointness
and
containment
in the interior.
l.
Introduction
The work reported here has been motivated by the practical need for a formal
understandingof spatialrelationswithin the realm of geographicinformation systems
(GIS). To display, processor analyzespatial information, usersselectdata from a GIS
by asking queries.Almost any GIS query is basedon spatial concepts.Many queries
explicitly incorporatespatialrelationsto describeconstraintsabout spatial objectsto
be analyzed or displayed. For example, a GIS user may ask the following query to
obtain information about the potentialrisksof toxic wastedumps to schoolchildrenin
a specific area: 'Retrieve all toxic waste dumps which are within 10 miles of an
elementary school and located in Penobscot County and its adjacent counties'. The
number o[elementaryschoolsknown to the information systemis restrictedby using
the formulation of constraints. Of particular interest are the spatial constraints
expressedby spatial relations such as within 10 miles, in, and adjacent.
The lack of a comprehensivetheory of spatial relations has been a major
impedimentto any GIS implementation.The problem is not only one of selectingthe
appropriateterminology for thesespatial relations,but also one of determiningtheir
semantics.The developmentof a theory of spatial relations is expectedto provide
answersto the following questions(Abler 1987):
o What are the fundamental geometric properties of geographic objects neededto
describe their relations?
o How can theserelations be defined formally in terms of fundamental geometric
properties?
o What is a minimal set of spatial relations?
0269 3798191
$3.00() l99l Taylor & FrancisLtd
162
M. J. Egenhoferantl R. D. Franzosa
psychological
In addition to the purely mathematicalaspects,cognitive,linguisticand
if
a
theory about
included
considerations(Talmy 1983,Herskovits 1986)must also be
(NCGIA
1989)'
developed
spatialrelations,applicableto real-world problems,is to be
have
which
concepts
Wittrin the scopeoi thlt paper, only the formal, mathematical
considered.
been partially provided from point-set topology will be
Any
rhe applicaiionof sucha theory of spatialrelationsexceedsthe domain of GIS'
formal
a
from
will
benefit
data
branch of scienceand engineeringthat dealswith spatial
logic and
understandingof spatiai relations.In particular,its contribution to spatial
computerengineering,
surveying
spatial,"uroning *ill ulro be helpfulin areassuchas
manufacturing (cAD/CAM), robotics and very largeaided design/computer-aided
scale integrated (VLSI) design.
(l)
The variety ofspatial relationscan be grouped into three differentcategories:
of
the
transformations
topological
topological relations which are invariant under
relationsin
..i"."ni" objects(Egenhofer1989,Egenhoferand Herring 1990);(2)metric
(3)
relations
and
1987);
Ci-Xiang
(Peuquet
and
terms of distancesand directions
by
as
described
(Kainz
1990)
objects
concerningthe partial and total order ofspatial
et
al'
Chang
1975,
(Freeman
below
and
prepositio-ns,u"h u, infront of, behind,aboue
spatial
topological
paper,
only
this
of
lggg, Hernendez 1991J.Within the scope
relationsare discussed.
oneFormalismsfor relations have so far been limited to simple data types in a
(Allen
intervals
as
e.g.
combinations,
or
their
dimensionalspacesuchas integers,reals,
in higher
l g83).Spatialdata, suchas geographicobjectsor cAD/CAM models,extend
spaceis
a
in
such
relations
primitive
of
set
a
dimcnsions.It has been assumedthat
systematically'
assumption
this
explore
to
made
richer,but so far no attempt has been
The goal of this paperis two-fold. First, to show that the descriptionof topological
is fairly
spatial rlelation,in teims of topologically invariant propertiesof point-sets
may
point-sets
two
between
relation
spatial
the topological
simple.As a consequence,
a
exists
there
that
show
to
Second,
effort.
be determinedwith little computational
that
state
not
does
This
falls.
relation
spatial
topological
any
lramework within which
may
the set of relations determined by this formalism is complete, i.e. humans
coverage,
complete
a
provides
formalism
the
that
but
relations,
distinguishadditional
relations
i.e. any such additional relation will be only a specializationof one of the
described.
The
As the underlying data model, subsetsof a topological spacewere selected.
topological
of
point-sct approach is the most generalmodel for the representation
using
spatial..glonr. Other approachesto the definitionof topologicalspatialrelations
complexes
simplicial
differentmodels,suchai intervals(Pullar and Egenhofer1988),or
(Egcnhofer1989),are gencralizedby this point-sct approach'
to
This paperis organizedas follows.The next sectionreviewspreviousapproaches
of
concepts
relevant
dcfiningiopological spatial relations.Section3 summarizesthe
paper'
the
of
remainder
point-sct topology and introducesthe notions used in the
shows their
Section4 introducesthe definition of topological spatial relations and
two
between
relations
the
realizationin R2. Section 5 investigatesthe existenceof
geographic
to
application
spatialregions,subsetsofa topologicalspacewith particular
compared'
data handling. In Section6, ihe relations within R (n)2) and R1 are
Previouswork
computer
Various collections of terms for spatial relations can be found in the
er a/'
Chang
1982'
Guptill
and
scienceand geographyliterature(Freeman1975,Claire
2.
19 8 9
Ingr
19 8 8
inn
und
whic
I
poln
adv
bee
defi
spa
'spa
top(
lOgl
ASS
forl
top
bee
ope
Th,
cor
In
tha
bo
fro
en
an
In
cn
or
Topologicalrelations
logical
'about
,1e8e).
h have
. SA
. ny
formal
;ic and
lputer'large:ies:( I )
of the
ionsin
lations
bed by
g et al.
spatial
a oner(Allen
higher
paceis
tically.
,logical
s fairly
[l'
uv
lxlstsa
Itethat
ls may
[erage,
llations
d. The
,logical
rsusing
rplexes
chesto
eptsof
paper.
's their
9n two
;raphic
:ed.
nputer
ryet al.
163
1989,Molenaar 1989).In particular,designsof spatial query languages(Frank 1982,
I n g r a m a n d P h i l l i p s 1 9 8 7 ,S m i t h e t a l . 1 9 8 7 ,H e r r i n g e t a l . 1 9 8 8 ,R o u s s o p o u l o e
st al.
1988)are a reservoirfor informal notationsof spatialrelationswith verbalexplanations
in natural language. A major drawback of these terms is the lack of a formal
underpinning, because their definitions are frequently based on other expressions
which are not exactly defined, but are assumedto be generally understood.
Most formal definitionsof spatial relationsdescribethem as the resultsof binary
point-set operations.The subsequentreview of these approacheswill show their
It will be obvious that none of the previous studieshas
advantagesand deficiencies.
beenperformedsystematicallyenoughto be usedas a meansto prove that the relations
defined provide a complete coveragefor the topological spatial relation between two
spatial objects. Some definitions consider only a limited subset of representationsof
'spatial
objects',whereasothers apply insufficientconceptsto definethe whole range of
topological spatial relations.
A formalism using the primitives distanceand directionin combination with the
l o g i c a l c o n n e c t o rA
sND,OR and NO7(Peuquet1986)willnotbeconsideredhere.The
assumption that every space has a metric is obviously too restrictive so that this
formalism cannot be applied in a purely topological setting.
The definitionsof relationsin termsof setoperationsusepure settheory to describe
topological relations. For example,the following definitions basedon point-sets have
been given for equal, not equal, inside, outside and intersecrs in terms of the set
operations :, #, c and n (Giiting 1988):
x : y : : p o i n t s( x ) : p o i n t s( y )
x * y '.: points (x) I points (y)
x i n s i d ey : :
p o i n t s( x ) c p o i n t s ( y )
x outside y :: points (x)npoints (il:
x i n t e r s e c t sy : :
A
p o i n t s( x ) n p o i n t s( i l + A
The drawback of thesedefinitionsis that this setof relationsis neither orthogonal nor
complete.For instance,equal and insideare both coveredby the definition of intersects.
In contrast,the model of point-setsper sedoesnot allow the definition of those relations
that are based on the distinction of particular parts of the point-setssuch as the
boundaryand the interior.For example,the relation intersects
is topologicallydifferent
from that where common boundary points exist,but no common interior points are
encountered.
The point-set approach has been augmentedwith the considerationof boundary
and interior so that ouerlapand neighborcan be distinguished(Pullar 1988):
x overlay y::
boundary (x)nboundary (y)lQ
and
interior (x)ointerior (i + A
x neighbor y::
boundary (x)nboundary (y)lQ
and
interior (x)ninterior (y) : b
In a more systematicapproach, boundaries and interiors have been identified as the
crucial descriptionsof polygonal intersections(Wagner 1988).By comparing whether
or not boundaries and interiors intersect,four relations have been identified: (l)
164
M. J. Egenhoferand R. D. Franzosa
(2) separation where
neighborhoodwhere boundaries intersect, but interiors do no[
the boundaries do
where
neither boundaries nor interiors intersect;(3) strict inclusion
boundaries and
both
with
not intersect, but the interiors do; and (4) intersection
the description
for
method
interiors intersecting.This approach usesa single,coherent
For
its
consequences'
all
in
out
of topological
*no spatial relations, but it is not carried
for
because
equality,
and
intersection
distinction can be made between
.*u-pl.,
both relations boundaries and interiors intersect'
Point-set topology
This model of topological spatial relations is based on the point-set topological
and results
notions of interiorandboindary. In this section the appropriate definitions
proofs'
without
stated
are
results
of
the
from point-set topology are presented.Some
found
be
can
and
definitions
of
the
Those proofs are all straightforward consequences
(1966).
Spanier
(1966)
and
Munkres
in mosi basic topology text books, e.g.
the
Let X be a set. A topology on X is a collection .q/ of subsetsof X that satisfies
arbitrary
under
is
closed
-ql
in
.{;(2)
X
are
three conditions:(l) the empty set and
spaceis a set X with
unions; and (3) -el is closedunder finite intersections. A topological
sets,and their
open
called.
are
X
on
topology
a topology il on X. The sets in a
(1)
contains the
sets:
closed
of
The
collection
sefs.
complementsin X are called closed
under finite
(3)is
closed
intersections;and
arbitrary
"-piy setand X;(2) is closedunder
3.
unions.
closenessis
Via the open sets in a topology on a set X, a set-theoreticnotion of
of x' This
neighborhood
be
a
to
is
said
u
then
established.Ifu is an open t"t und xeu,
A
metric d
closeness'
of
notion
metric
the
generalizes
set-theoreticnotion of c^loseness
d. This
by
defined
topology
metric
the
x,
called
on
on a set X induces a topology
the
that
c
>
0
such
is
an
there
x€u,
each
il
for
set
open
X
is
an
topology is suchthat u
distance
whose
points
of
set
is
the
d-ball
A
in
U.
d-ball of radius s around x is contained
from x in the metric d is lessthan s, i.e' {yeXld(x,y)<e}'
subsetof X
For the remainderof this paperlet X bsa setwith a topology -&.lf s is a
and is
topology
subspace
the
is
called
topology
This
.il
.
from
then S inherits a topology
if
U:Snlzfor
only
and
if,
topology
subspace
in
the
open
definedsuch that U-iis
X.
some set ve,il.lJnder such circumstances,S is called a subspaceof
3.7. Interior
all open
Given Yc X,the interior of X denotedby Y', is definedto be the union of
in Y
contained
set
open
largest
the
Yis
of
interior
i.e.
the
in
setsthat are contained {
in
y
contained
X i'e'
y is in the interior of Y if and only if there is a neighborhood of
'yeY"
could
lf,and only if, thereis an open set U suchthat ye U c Y' The interior of a set
is X itself. If U is
t" "'npty, e.g.the interior of the empty set is empty. The interior of X
o p e n t h e n U " : U . l f Z c - Y t h e nZ " c Y " '
3.2. Closure
The closureof y, denoted by X is definedto be the intersection o[ all closedsetsthat
that y is
contain { i.e.the closureof Yis the smallestclosedsetcontaining X It follows
y
e
rif and
y
i
n
t
e
r
s
e
c
t
s
{i.e'
intheclosureof Yif andonlyif everyneighborhoodof
with
set
only
is
the
set
y.
emp-ty
The
only if U oY * Q lor everyopen set U containing
:
Z
c
Y
t
h
e
n
Z
c-Y'
C
.
l
f
. - p t y c l o s u r e . T h e c l o s u r e o i X i t X i t s e l f . IC
f isclosedthene
3.3.
clos
follc
bott
eve
X a
1
A
J.+.
disc
inte
bec
the
foll
iss
tha
xit
an(
3.
for
pa
(2)
sa
em
col
cor
thr
sh
or
C
e
er
co
in
Topological relations
n where
ariesdo
'ies and
cription
;es.For
rusefor
rlogical
I results
proofs.
e found
ifiesthe
rbitrary
lXwith
td their
ainsthe
lerfinite
ienessls
f x. This
retricd
d. This
thatthe
listance
setofX
y andis
inV for
allopen
redin Y
in X i.e.
etcould
ilf U is
etsthat
haty is
f ll ancl
ietwith
tZcY.
165
3.3. Boundary
The boundary of Y denoted by 0Y is the intersection of the closure of Y and the
closure of the complementof X i.e. fif :ln-y -y. The boundary is a closed set. It
follows that y is in the boundary of Y if and only if every neighborhood of y intersects
both Yand its complement,i.e. ye?Ylf and only 1f UaY*b
and Un(X- Y)*Q for
every open set U containing y. The boundary can be empty, e.g.the boundaries ofboth
X and the empty set are empty.
3.4. Relationshipbetweeninterior, closure and boundary
The conceptsof interior, closure and boundary are fundamental to the forthcoming
discussionsof topological spatial relations between sets.The relationships between
interior, closure and boundary are described by the following propositions:
Proposition3.1. Y" a?Y: Q.
Proof: lf xe?Y then every neighborhood U of x intersectsX - Y so that U cannot
be contained in Y As no neighborhood U of x is contained in Yit follows that x{ Y" and,
therefore,
1YaY": A.
T
Proposition
3.2. Y"v0Y:Y.
ProoJ': Y" cYc Yand, by definition,0YcY. As Y'and AY areboth subsetsof 7it
followsthat (Y'vdY)c 7. To show that Ic(Y"udY),let xeland assumethat x{Y". It
is shown that x€rywhich. sincexe-Xonly requiresshowingthat xeX - Y.r{Y" implies
that every neighborhood ofx is not contained in { therefore,every neighborhood of
x intersectsX - Y, implying that xeX - Y So xed Y. Thus if xe Y and r{ Y', then xed {
a n d i t f o l l o w st h a t 7 c ( Y " v 0 Y . t h u s
l:(Y"vAY).
tr
3.5. Separation
The concepts of separation and connectednessare crucial for establishing the
forthcoming topological spatial relations betweensets.Let Y c X . A separationof Yis a
pair A, -Bof subsetsof X satisfyingthe following three conditions: (1) A+ Q and B + O;
(2) Av B : Y; and (3) A a B : Q and A o B : Q. If thereexistsa separation of I then Y is
saidto be disconnected,
otherwiseYis said tobe connected.lfY ts the union of two nonempty disjoint open subsets of X, then it follows that Y is disconnected. If C is
connectedandCcD c e, then D isconnected.In particular,if Cis connected,then f is
connected:however. AC and C" need not be connected.
Proposition 3.3. lf A, B form a separation of Y and if Z is a connectedsubsetof {
t h e n e i t h e rZ c A o r Z c B .
Proof: By assumption,Z is a subsetof the union of A and B,i.e. Z c,4uB. It is
shown that the intersection betweenZ and one of A or B is empty, i.e.either Z oB : Q
or ZnA:gj.
Supposenot, i.e. assumethat both intersectionsare non-empty. Let
C:ZaA
and D:ZaB.
Then C and D are both non-emptyand CvD:2.
As
(becauseA, B ts a separation of Y), it follows that
e -A, DcB, and AoB:@
eaD:Q.
Simllarly, CaD:Q;
therefore, C and D form a separation of Z,
contradictingthe assumptionthat Z is connected.So either ZoB:@
or ZoA:6,
i m p l y i n gt h a t e i t h e r Z c A o r Z c B .
T
166
M. J. Egenhoferand R. D. Franzosa
The following
A subset Z of X is said to separate X 1f X Z is disconnected'
of
a subset of X
separation result gives simple conditions under which the boundary
separatesx.
p r o p o s i t i o3
n . 4 . A s s u m eY c X . l f Y " + o a n d Y + X , t h e n Y ' a n d X - Y f o r m a
separation of X - i.Y, and thus d Y separatesX'
side
com
lnva
T
spat
prooJ.. By assumption,Yo and X -7 arc non-empty. Clearly, they are disjoint
t t f o l l o w st h a t Y ' a n d
o p " n r " i r . p r o p o s i t i o n3 . 2 i m p l i e s t h a t X - A Y : Y " v ( X - f .
tl
0Y.
X- Yform a separationof X
sum
I
3.6. T opologicalequiualence
Two
The study of topological equivalenceis central to the theory of topology'
topological
same
the
or
of
(homeomorphic
topological spacesarc topologicallyequiualent
bijective
typel lt theie is a bijective function between them that yields a
function,
a
Such
topologies.
respective
in
the
sets
open
"or.lrponA"nce between the
Examplesof
which is continuouswrth a continuousinverse,is calledahomeomorphism.
and skew'
scale
rotation,
of
translation,
notions
Euclidean
homeomorphismsare the
are called
homeomorphism
under
preserved
are
that
spaces
Properties of topological
is a
connectedness
property
of
the
example,
For
spaces.
thi
of
topitogical inuariants
topological invariant.
A framework for the descriptionof topological spatial relations
This model describingthe topological spatial relations betweentwo subsets,,4 and
of the
B, of a topologicalspaceX is basedon a considerationof the four intersections
and
boundariesand interiors of the two sets A and B,i.e. i Aa1B, A"oB" aAoB"
A"nAB.
4.
DeJtnition4.1. LeLA, B be a pair of subsetsof a topologicalspaceX. A topological
spatial relation between A and B is describedby a four-tuple of values of topological
A"ai B'
invariants associatedto each of the four sets d,4ndB, A".. B", AAaB", and
respectivelY.
A topologicalspatialrelationbetweentwo setsis preservedunder homeomorphism
A, B c X,
of the underlyingspaceX. Specifically,lff : { + Y is a homeomorphismand
onto
then 0A\iB, A''a8", i AnB", and A"ai B are mapped homeomorphically
the
Since
Af@)oAfGi), f@)"qf(B\', Af@)af(B)', and f($"oAf@), respectivelv'
of
these
invariants
topological spatial relation is defined in terms of topological
in X is
iniersections,it follows that the topological spatial relation between A and B
Y'
tn
identical to the topological spatial relation between f(A) and f(B)
A topological ipatial relation is denoted here by a four-tuple ( ' -' ' )' The
to the four
entriescorrespondin order to the valuesoftopologicalinvariantsassociated
intersection,
boundary
The first intersectionis called the boundary
set-intersections.
the
the second intersection the interior-interior intersection, the third intersection
houndary
interior
the
houndary interior intersection, and the fourth intersection
intersection.
4.1. Topological spatialrelutionsfrom emptyfnon-emptyset-interse(tions
As the entries in the four-tuple, properties of sets that are invariant under
homeomorphismsare considered.For example,the propertiesempty and non-empty
FA
spat
ther
setc
topo
exac
I
on v
exls
poin
4.2.
I
in th
A a
con
in th
equ
and
l s l
Tab
Topological relations
lowing
etofX
form a
lisjoint
Y" and
n
i. Two
'logical
rjective
nctlon,
rplesof
lskew.
r called
issis a
,Aand
l ofthe
B" and
,logical
,logical
l"niB,
are set-theoretic,and therefore topologically invariant. Other invariants, not considered in this paper, are the dimension of a set and the number of connected
components(Munkres 1966).Empty/non-empty is the simplest and most general
invariant so that any other invariant may be considereda more restrictiveclassifier.
For the remainder of this paper, attention is restrictedto the binary topological
spatial refations definedby assigningthe appropriate value of empty(e) andnron-empty
( 1O) to the entriesin the four-tuple.The 16 possibilitiesfrom thesecombinations
are
s u m m a r i z c di n t a b l c l .
A set is either empty or non-empty;therefore,it is clear that these l6 topological
spatialrelationsprovide completecoverage,that is,givenany pair of setsA andB jn X,
thereis alwaysa topologicalspatialrelation associatedwith A and B. Furthermore,a
set cannot simultaneouslybe empty and non-empty,from which follows that the l6
topologicalspatialrelationsare mutually exclusive,i.e.for any pair of setsA andB in X,
exactly one of the l6 topological spatial relations holds true.
In general,eachof the l6 spatialrelationscan occur betweentwo sets.Depending
on variousrestrictionson the setsand the underlyingtopologicalspace,the actualset
o1
existingtopologicalspatial relationsmay be a subsetof the 16 in table l. For general
point-setsin the plane R2, all 16 topologicalspatialrelationscan be realized(figure
1).
4.2. InfluenceoJ'the topologlicalspaceon the relations
The setting,i.e.the topologicalspacex in which A and Blje,playsan important role
in thc spatial relation betweenA and B. For example,in figure }(leit panel) the
two sets
A and t have the relation (a,re,re,-tb)
a s s u b s e t so f t h e l i n e . T h e s a m e
configurationshowsa differentrelation betweenthe two setswhen they are embedded
in the plane(figure2, right panel).As subsetsof the plane,the boundariesof ,4 and
B are
equalto A and B, respectively,
and the interiorsare empty, i.e.AA: A, A,.: qj, AB: B,
and B" : g-ltfollows that in the planethe spatialrelationbetweenthe two sets.4
and.B
] sF b , b , Q , A )
Table l.
rphism
Bc X,
F Onto
ce the
I these
inXis
_).The
hefour
ectlon,
on the
undary
under
-emPt)l
167
The I 6 specifications
ofbinary topologicalrelationsbasedon the criteriaofemptv and
non-empty intersectionsof boundariesand interiors.
"n0
i,a"
r
rt
1
11
1
rs
1
r't
1
rs
rro
trr
rrz
rr:
tr+
rts
o
A
2
lQ
b
4
lA
A
6
lQ
A
1b
8
A
-lg
A
1A
A
lb
A
lb
a
rx
a
a
YJ
tg
1@
a
a
1A
1A
@
g
1a
.a
@
V)
fb
1A
l
t
a
o
a
g
b
q
Q a
a b
t
g
Q
t
@
g
b ) o
a t a
a t -ta
a
a
)q
ta
lzi
)Q
)b
ta
tg
tA
M. J. Egenhoferand R. D. Ftanzosa
r68
r5
r2
lo
ASSUN
N
@
gua12
^'@ B=O
non-
n'!
l(xx)
@ B = o ^ =@ B ' O
A=/t
B= $
T
D
It
Furtl
set.F
cond
beca
r4
s
A=-
5 . 7
It
polyg
cons
B=o
B-O
@ B'O
@ B'O
rto
t ^
beca
need
T
emp
I
tl//
I
Fror
v
aA
A= @ a=I t 2
A =@ B = -
^ =@ B = O ^ =@ B =O
r14
r t J
rr5
N
^ = @B
l =-O ^ = 6
Figure l.
B-S
^=@B=o
@ B=O
of
Examples of the l6 binary topological spatial relations based on the comparison
interiors'
and
boundaries
between
empty ind non-empty set-intersections
i
andB with (leftpanel)thetopologicalspalial
Figure2. Thesameconfigurationof thetwo sets,4.
relatio@
n ,-t@,-tb,t@)whenembeddedinalineand(rightpanel)(l@'g'A'@)ina
plane.
conl
5 . 1. .
regi
two
rela
folk
rela
and
bou
salT
rela
bot
Topologlicalrelations
169
5. Topological relations betweenspatial regions
It is the aim of this paper to model topologicalspatialrelationsthat occur between
polygonal areas in the plane; therefore, the topological space X and the sets under
considerationin X are restricted.Theserestrictionsare not too specificand the only
assumptionthat is made about the topological spaceX is that it is connected.This
guaranteesthat the boundary of each set of interestis not empty.
The setsof interestare the spatial regions,definedas follows:
Definition5.1. Let X be a connectedtopologicalspace.A spatial region in X is a
non-empty proper subsetA of X satisfying(1) ,4" is connectedand (2) A:d'.
It follows from the definition that the interior of each spatial region is non-empty.
Furthermore,a spatialregionis closedand connectedas it is the closureof a connected
set.Figure 3 depictssetsin the plane which, by failing to satisfyeither condition (1)or
condition(2) in Definition 5.1,are not spatialregions.A and B are not spatialregions,
becauseA' and Bo are not connected,respectively.c and D are not spatial regions,
becausethey fail to satisfy condition (2), i.e. c + c" and D + a . rne latter sets are
neededto realizethe topologicalspatialrelationsr2,r5,ts1re,r12?nd r,. in the plane.
The following proposition impliesthat the boundary of eachspatialregion is nonempty.
Proposition5.2. If .4 is a spatial region in X then AA+A.
Proof: A" +g.A:
A s i n c e , 4i s c l o s e d a
, n dA * X b y d e f i n i t i o n o fa s p a t i a lr e g i o n .
From proposition 3.4 it follows that A" and X -/ form a separationof X-aA.rf
AA:Q then the two sets form a separationof X, which is impossiblesince X is
tr
connected;
therefore,AA + 6.
5.1. Existenceof region relations
The framework for the spatial relations between point-sets carries over to spatial
regions,however, not all of the 16 relations betweenarbitrary point-setsexist between
two spatial regions. From the examples in figure I it is concluded that at least the
relationsrg, r1, r3t 16111, r16 r1r, t14 and rrc exist betweentwo spatial regions.The
following proposition shows that thesenine topological spatial relations are the only
relations that can occur between spatial regions.
Lparison
of
Proposition5.3. For two spatial regionsthe spatial relations t2, r41r51rs,rslty2
and rr. cannot occur.
Proof: This begins by proving that if the boundary interior or interior
boundary intersectionis non-empty then the interior-interior intersection betweenthe
same two regions is also non-empty. This implies that the six topological spatial
relationst4, t51 rg; rs, trz and r13, all with empty interior interior and non-empty
boundary-interioror interior boundary intersections.cannot occur.
W
icalspatial
(V (l\in
e
(c)
Figure 3.
@
Sets in the plane that are not spatial regions.
-
(D)
M. J. Egenhoferand R. D. Franzosu
ll\
Lct ,4 and B be spatialregionsfor which 0AoB' + b.lt is shown that A" ^8" + Q.
so A wiA
A:l:1.
U s i n g p r o p o s i l i o n3 . 2 . A ' t t A A : A a n d A w l ( A l : A
A
"
o
AA:q'h
a
n
d
A
.
o
0
(
A
.
)
:
Q
3
.
1
,
: A ' v A ( A ' ) . F u r t h e r m o r eb, y p r o p o s i t i o n
followsthat0(1"):d.4.NowletxeAAoB",thenxe1i,/ '),andsinceB"isopenand
i f t h e b o u n d a r y i n t e r i o r i n t e r s e c t i o ni s
c o n t a i n s- x .i t f o l l o w st h a t . 4 " n B ' l Q . T h u s
is also non-empty. It also follows
intersection
non-empty,then the interior interior
then the interior interior
is
non-empty,
that if the interior,boundary intersection
intersectionis also non-emPtY.
Next rt is proved that if the boundary boundary intersectionis empty and the
interior interior intersectionis non-empty,then either the boundary interior or the
interior boundary intersectionis non-empty.This impliesthat the spatial relation rr'
with a non-emptyinterior interior intersectionand empty intersectionsfor boundary
boundary,boundary-interiorand interior boundary,cannot occur.This will complete
the proof of the proPosition.
Let A andB be spatial regionssuchthat AAoAB: Q and A" nB" + A.Itis shown
thatif0AoB":Ej,ther.A'oi B*/.AssumethatAAoB":gj'SinceB:B"vAB'it
followsthati.AoB:Qand,therefore,Bc-X-SA.Proposition3'4impliesthat'4'and
X - A form a separationof x -aA, and since-Bis connected,proposition 3.3 implies
AO
SC
Pr
AO
A<
B
Fu
thateitherBcA'-'orBcX_ A.Since,byassumption,A''oB"+Q,iLfollowsthat
I
B r: A' and, therefore, AB c,4'. Clearly, 0BoA' * Q and the result follows'
5.2. Semanticsof region relations
In figure 1,examplesweredepictedfor the topologicalspatialrelationsts, r1, t3, 16.
11, rto, r11, ryq and r,, between spatial regions' Each of these nine relations is
"onriJ"."d in the definitions below and their semanticsare investigatedusing the same
notation as in Egenhofer(1989)and Egenhoferand Herring (1990)'
Definition 5.4. The descriptiveterms for the nine topological spatial relations
betweentwo regionsare given in table 2.
If the topological spatial relation betweenA and B is ro then, in the set-theoretic
sense,.4 and B are disjoint and, therefore, the topological spatial relation disjoint
coincideswith the set-theoreticnotion of disjoint. The following proposition and
corollariesjustify the other descriptiveterms for the topological spatial relations
definedin table 2.
the
co
B
A=
Tc
un
oc
Et
Table 2. Terminology used for the nine relations between two spatial regions.
dn"
ino'
ro
\9,
11
llQ,
13
FA,
16
11
rr,
rrr
rr+
tts
@'
FA,
@,
FA,
(q,
F@,
b,
@,
)4,
14,
b,
)q,
)@,
14,
)4,
19,
)4,
1Q,
a,
a,
a,
a,
t@,
14,
''nO
6\
6
A)
6
6
16)
)A)
-lA)
)A)
SU
pc
A and B are disjoint
,4 and B touch
.4 equals B
,4 is inside of B or B contains .4
,4 is covered bY B or B covers ,4
.4 contains B or B is inside of .4
.4 coversB or B is coveredbY I
A and B overlap with disjoint boundaries
A and B overlap with intersecting boundaries
th
th
sp
X
is
re
or
pr
Topological relations
lll
'oB'*Q.
o A"voiA
11:Q. tt
openand
rsectionis
so follows
rr interior
y and the
jor or the
elationrr,
roundary
lcomplete
t is shown
B"uiB, it
nI A" and
1.3implies
llowsthat
.
!
Proposition 5.5. Let A and B be spatial regions in X. If A"oB"*e
A " o A B : Q , t h e n A " c B " a n dA c - B .
and
Proof: .4'is connected. proposition 3.4 implies that B" and X-B form
a
separationof X. SinceA" oaB: Z, it follows by proposition 3.1that A,.c B"v(X _
B).
P r o p o s i t i o n 3 . 3 i m p l i e s t h a t e i t h eAr" c : 8 " o r A " c - ( X * B ) . B u t A " o B " l e ; t h e r c f o r e ,
A o c B . S i n c eA c - 8 " . i t f o l l o w st h a t A ' c B ' w h i c h . b y d e f i n i t i o n5 .l . i m p l i e s
that
Ac-B.
U
From proposition 5.5 it follows that if A is covered by B, then ,4 c
B; therefore,the
spatialrelationis coueredby coincideswith the set-theoreticnotion
of beinga subsetof.
The following corollary to proposition 5.5 shows that the
spatial relationequal
correspondsto the set-theoreticnotion of equality.
Corollary 5.6. Let A and B be spatial regions.If the spatial relation between
,4 and
B is r., thenA:8.
Proof: A" aB" * b and A" ai B: @;therefore,proposition 5.5implies that A c_
B.
F u r t h e r m o r eA, A a B " : Q . A g a i nb y p r o p o s i t i o n5 . 5 , 8 c , 4 . T h u s , 4 : 8 .
I
The following corollary to proposition 5.5 showsthat if ,4 is insideB, then A c, 8,,:
therefore' the spatial relation inside coincideswith the topological notion of being
containedin the interior. Conversely,containscorrespondsto containsin the interior.
;rf1,f31t6,
rlationsis
! thesame
relations
rtheoretic
n disjoint
ition and
relations
anes
undaries
Corollary 5.7. Let A and B be spatial regions.If the spatial relation between
,4 and
B is ru, then ,4 c B'-.
Proof: Proposition 5.5 implies that A,' cB" and Ac_8. By proposition
3.2,
A:A"wAA and B: B"wAB.So il.4cB. Since i Aoi B:e,
i t f o l l o w st h a t d , 4 c B " .
Togetherwith .4'cB' this implies Ihat Ac_B".
I I
6. Relationsin n-dimensionalspaces
It is natural to ask 'What further restrictionson the topological spaceX and the
sets
under considerationin X further reduce the topological spatial relations
that can
occur?'This section will explore this question by consideringthe casewhere
X is a
Euclideanspace.
R' denotesn-dimensionalEuclidean space with the usual Euclidean metric.
A
subsetof R" is boundel if there is an upper bound to the distancesbetweenpairs
of
points in the set; otherwise,it is said to be unboundetl.
The unit disk in R' is the set ol points in Rn whosedistancefrom the origin
is less
than' or equal to, 1. The unit spherein R' is the set of points in Rn whose distance
from
the origin is equal to 1. For n>l the unit disk in Riis connected.For n>-2 the
unit
spherein R' is connected.Let x be a topological space.An n-diskin x is a subspace
of
X that is homeomorphicto the unit disk in R,. An n-spherein X is a subspaceof X
that
is homeomorphicto the unit spherein Rn* 1.n-disksin Rnare boundedand are
spatial
regions;the latter is a relatively straightforward consequenceof the Brouwer theorem
on the invarianceof domain (Spanier 1966).Since n-disks in Rn are spatial regions,
proposition5.3 restrictsthe number of spatial relationsthat can occur
betweenthem.
t72
M. J. Egenhoferand R- D' Franzosa
Inproposition6.litisshownthatifAandBarezr-disksinR,withn22,thenthe
cannot occur. The proof of this
,pu;i;i reiation ouerlap-with disjoint boundary
proposition is based on the following two facts:
is an (n- l)-spherein Rn and'
Fact 1. Let Abean n-disk in R'with n22'Thend'4
therefore,connected.
Thisfact,also,isaconsequenceoftheBrouwertheoremontheinvarianceofdomain
(Spanier 1966).
Fact 2. Let ,4 be an n-diskin R'with n)2'Then
unbounded.
R"-A"
i s c o n n e c t e da n d
Thissecondfactisa(non-)separationtheoremrelatedtotheJordanBrouwer
separation theorem (Spanier 1966)'
Proposition6,|.Thetopologicalspatialrelationrl4'overlapwithdisjoint
with n>2'
boundaries,does not occur bitween rc-disksin R'
shown that if 0 A oa B : a, then
Pr oof., Let A andB be n-disksin R, with n) 2.|tis
relation ouerlap with disioint
A and B do not ou"ilup and, therefore, the spatial
boundariescannot occur.
wr]l^bederived'B.is a
Assume 0Aai,B:A and / and B overlap' A contradictignB" and R" B form a separatlon
spatiat region; therefore,proposition 3.4 implies that
By fact l' AA is connected'
i.AcR"-08'
that
it totto*t
of R'-dB. As i,AaLB:'f
SinceA and B
1Ac(R"-B).
or
aAcB"
either
that
therefore,proposition 3.3implies
AAcB"'
overlap,iitotio*t that 0AoB" *O and' therefore'
. sing
f
a
c
t 2 , R " - B : i s c o n n e c t e dU
0AcB" impliesthat d.An(R;-8"):Q.By
.4' or
(R'B')c
that either
proporition, 3.i and l.a una aiguing as above, it follows
(R"-8.)c(R^_ A)'Thefirstcaseyieldsacontradictionbecause'byfact2,R"-8"
i s u n b o u n d e d , b u t , 4 " i s n o t . T h e s e c o n d c a s e i m p l i e s t h a t A c B " a n d , t h e r e fino r e ,
the assumptionthat A and B overlap' Therefore'
A"ai;B:Q,whichcontradicts
that the spatial relation r14 cannot
either casea contradiction is obtained and it follows
fl
occur betweenn-disksin R" with n)2'
o\€tlap with intersecting
f
Note that tor n.>2 the topological spatial relation 15,
l)'
(figure
boundaries,does occur between two n-disks
between1-disks,while rt t'
The oppositerituiiion o""uis in R1 where'r,-ncan occur
that r rncan occur between two
ouerlapwith intersectingboundaries,cannot. It is clear
cannot occur' Its proof requires
1-disksin R'(figure 2)."Proposition6.2 showsthat rt,
is either a closed interval lq'b7 for
rhe easily derived taci that'a spatial region in Rr
for some ceRl'
,so-" o,b.Rt, or a closed ray la,co) or (-co'a]
Proposition6.2.Thetopologicalspatialrelationrt'doesnotoccurbetween
s p a t i a lr e g i o n si n R 1 .
It is
proof: Let ,4 and B be spatial regionsin R1 and assumethat '4 and B overlap'
interval or a closedray;therefore'
shown that 0Aa0B: O-Eainof 'q aidB is a closed
the others can be proven
selected;
is
One
there are nine differeni casesto examine.
accordinglY.
T h e n A A : { a } a n d A B : { b } ' S i n c eA a n d B
A s s u m eA : l a , o o ) a n d B : ( - m , b l '
tr
overlap,it follows thal a<b,*tti"tt i-pti"s that d'4ndB--Q'
7. Cr
AI
is bas
distan
the bc
and nr
beeni
topok
relati<
AI
imme
syste
be prr
spati
for in,
have
exten
topol
inters
and I
for fc
T
nece
dime
and 1
plan
are I
betw
fram
such
Ack;
with
Bart
part
Cor
gra
No.
109
Ref
Asr
Au
CH
Cr
Topological relations
2, thenthe
of of this
in R'and,
of domain
ected and
-Brouwer
h disjoint
: gj,then
'h disjoint
red.B is a
eparatron
onnected,
>AandB
ed.Using
i o ) c , 4 oo r
2, Rn-Bo
therefore,
lrefore,in
14cannot
n
tersecttng
w h i l er , . ,
weentwo
f requires
la,bf for
: between
erlap.It is
therefore,
)e proven
AandB
n
t73
7. Conclusion
A framework for the definition of topological spatial relations has beenpresented.It
is basedon purely topological properties and is thus independent ofthe existenceofa
distancefunction. Ihe topological relations are describedby the four intersectionsof
the boundaries and interiors of two point-sets.Considering the binary values empty
and non-empty for theseintersections,a set of l6 mutually exclusivespecificationshas
beenidentified. Fewer relations exist if particular restrictions on the point-setsand the
topological spaceare made. It was proved that there are only nine topological spatial
relationsbetweenpoint-setswhich are homeomorphic to polygonal areasin the plane.
Although the nature of this work is rather theoretical, the framework has an
immediate effect on the design and implementation of geographic information
systems.Previously,for every topological spatial relation a separateprocedure had to
be programmed and no mechanism existed to assurecompleteness.Now, topological
spatial relations can be derived from a single,consistent model and no programming
for individual relations will be necessary.Prototype implementationsof this framework
have been designed and partially impr:mented (Egenhofer 1989), and various
extensionsto the framework have been investigated to provide more details about
topological spatial relations, such as the consideration of the dimensions of the
intersectionsand of the number of disconnectedsubsetsin the intersections(Egenhofer
and Herring 1990).Ongoing investigationsfocus on the application of this framework
for formal reasoning about combinations of topological spatial relationships.
The framework presented is considered a start and further investigations are
necessaryto verify its suitability. Here, only topological spatial relations with codimensionzero were considered,i.e. the differencebetween the dimension of the space
and the dimension of the embeddedspatial objects is zero, e.g.between regions in the
plane and intervals on the one-dimensionalline. Also of interest for GIS applications
are the topological spatial relationships with co-dimension greater than zero, e.g.
betweentwo lines in the plane (Herring 1991).Likewise, the applicability of this
framework to topological spatial relations between objects of different dimensions,
such as a region and a line, must be tested.
Acknowledgments
The motivation for this work was given by Bruce Palmer. During many discussions
with John Herring, these concepts have been clarified. Andrew Frank and Renato
Barrera made valuable comments on an earlier version of this paper. This work was
partially funded by grants from NSF under No. IST 86-09123,Digital Equipment
Corporation under SponsoredResearchAgreementNo. 414 and TP-765536,Intergraph Corporation, and the Bureau of the Census under Joint Statistical Agreement
No. 89-23.Additional support from NSF for the NCGIA under grant number SES8810917is gratefullyacknowledged.
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relafions
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