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{F
m
,FO
ml
}
{F
l
,F
m
,F
r
}
{F
l
,FO
l
}
{F
l
,FO
l
,D
l
}
{FO
l
,FO
ml
}
{F
l
,F
m
}
{F
l
,FO
l
,F
m
,FO
ml
}
{FC
l
,FO
ml
}
{C
l
,FO
l
}
{D
l
,C
l
,FO
l
,BO
l
}
{FC
l
,C
l
}
{D
l
,FO
l
}
{FO
ml
,FC
l
,FO
l
,C
l
}
Fig. 2.
Singular relations defined by sets of general relations
Accordingly
would seem to be quite an appropriate descrip-
tion of what we really know about two parallel lines which are
{D
l
,C
l
,FO
l
,BO
l
}
probably
equal in
length.
Fig. 2 shows how singular relations are represented by sets of general rela-
tions. Only a quarter of all relations are depicted, since the other relations are
symmetrical to those in Fig. 2. As with the disconnection relations of
,only
disconnected singularities are considered.
Apparently connected singularities
are
treated as
apparently connected general relations
, i.e. they are conceived as dis-
connected relations in which distances become arbitrarily short. Our knowledge
gets more uncertain near singular relations - this uncertainty is represented by
sets comprising a number of possible relations rather than only one relation.
In particular, if two endpoints are in singular positions then these sets consist
of three or four general relations, depending on whether the endpoints lie on
the same singularity, e.g.
BA
{F
l
,F
m
,F
r
}
in Fig. 2, or on different singularities, e.g.
{D
l
,C
l
,FO
l
,BO
l
}
. By contrast, if there is only one endpoint in singular position
the sets consist of only two general relations. We observe that all singularities
are uniquely identified by this technique.
2.2
Reasoning with Singular Relations
How does this representation of singular relations affects reasoning processes?
Let us consider the example in Fig. 3. We assume that we know the relations
between
x
and
y
as well as those between
y
and
z
. For the position of
y
with
respect to
x
we write
x
y
, and accordingly we write
y
z
for the position of
z
with respect to
, i.e.
x
z
. We do this by the composition operation which was defined in [3]: for each
pair of general relations the transitivity relation is given. The left hand side of
Fig. 3 shows
y
. Our goal is to infer the relationship between
z
and
x
x
y
in singular relation; the composition result is indeterminate. In
comparison, the right hand side of Fig. 3 shows
x
y
in general relation; here the
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