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same radius, their relations can only be the situations listed in subset (2). The
relations between a disk with radius R and a column with radius R/2 can only be
the three situations listed in subset (6).
It is worth notice that subset (3) and (5) are respectively the reverse set of
subset (4) and (5). That is if
are two space area, their shapes determine
that the set composed of their possible relations R(
x
and
y
x,y
) is subset (2). Then when
we take into account the relation R(
), the result set is subset (3). So we can
believe that from (1) to (2), (3), then (4), (5), finally (6) reflects the relation of
shapes of the two geometric areas from specialization to common.
Then we can make two shape-changeless areas move in the space and change
their relations. The change is only restricted to the following four sequences:
(1) DC
ř
EC
ř
PO
ř
=
(2) DC
ř
EC
ř
PO
ř
TPP(TPP
-1
)
(3) DC
ř
EC
ř
PO
ř
TPP(TPP
-1
)
ř
NTPP(NTPP
-1
)
(4) DC
ř
EC
ř
PO
y,x
4.7.2
Temporal spatial relation
1. Position state and motion state
The eight RCC relations are classified into position state and motion state by
Galton. He has given the classification definition with some logical forms about
temporal relation by Allen. First we introduce the concepts, predicates, and
functions used by Calton.
(1) The described time is divided into interval and moment.
(2) The described predicate of state exists:
Holds-on(
s,i
) indicates state
s
exists on interval
i
.
Holds-at(
s,t
) indicates state
s
exists at moment
t
.
(3) Predicate Div(
t,i
) indicates moment
t
is in interval
i
.
(4) Function inf(
i
) indicates the beginning moment of interval
i
.
(5) Function sup(
i
) indicates the ending moment of interval
i
.
Definition 4.4
Position state: if state s satisfies
∀
i
(Holds-on(
s,i
))
ŗ
Holds-at(
s
,inf(
i
))
∧
Holds-at(
s
,sup(
i
))
