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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

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l

FC
r

D
l

C
l

C
r

D
r

I

b

Contains

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l

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r

BO
l

BO
r

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ml

BO
mr

Overlaps

Back

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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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