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a
left
right
middle
Neighbourhood Graph
F
l
F
m
F
r
During
FO
ml
FO
mr
FO
l
FO
r
Overlaps
Front
FC
l
FC
r
D
l
C
l
C
r
D
r
I
b
Contains
BC
l
BC
r
BO
l
BO
r
BO
ml
BO
mr
Overlaps
Back
B
l
B
m
B
r
Fig. 1.
Left: Interval relations embedded in two dimensions; the vertical reference in-
terval is displayed bold. Middle: A mnemonic description; Right: Singular relations
in which intervals are precisely aligned with each other. But in
there are
no such singular relations explicitly defined. For instance, there is no relation
between
BA
FO
l
which would correspond to an interval in which one end-
point is located exactly level with an endpoint of the other interval (see Fig.
1.(a)). The question arises as to how we deal with such singular arrangements,
in which one endpoint lies precisely at a location which marks the transition
between qualitative concepts, as between
F
l
and
FO
l
.Thisisimportant,since
no possible arrangement of intervals should remain undefined. This is the issue
we are interested in.
F
l
and
2
Singular Relations
First of all we show why singular relations exist at all. A qualitative repre-
sentation is the result of an abstraction process, which can be regarded as the
partitioning of a continuous space into a number of equivalence classes. As a side-
effect of this, singularities emerge as transitions between neighbouring classes.
For example, the continuous space of interval arrangements in
2
can be con-
ceived as consisting of all metrically distinct interval arrangements. A special
abstraction of this continuous space distinguishes the relations of
R
,inwhich
each equivalence class is a binary relation between two intervals. In the neigh-
bourhood graph in Fig. 1, the transition between two neighbouring classes marks
a singularity, for instance, between
BA
FO
l
. We refer to the interval relations
which fall into these transitions as singular relations. Fig. 1.(a) serves as an ex-
ample. An arbitrary small change in the position of one interval which forms a
singular relation with another interval transforms it in most cases into a general
relation. Such a small change applied to a general relation would not normally
change this relation.
We would like to argue that singular relations should not have the status of
basic relations in a qualitative representation. By contrast to
F
l
and
BA
,whichcom-
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