Object is a
on points of
is a circular
of thc Body.
returns thc
t consists of
returned. In
t-sct. which
cas or bodics
dies
The interior
as A?-A-
idaries and
nted by thc
= A°ml3°
nto account
|.2,3.4) can
4) have Ni
)etween the
: N possible
results in a
the objects
iis triplet is
the and(^),
> definition
).dim(B?))
-dim (B?))
lationships
seen in Fig. 5.
4 À
ee @ d
(a) A touch B (b) A in B (c) A overlap B
4 À
eue das
(c) A disjoint B
(d) A cross B
Fig. 5. Some Examplc
The five relationships are mutually exclusive, that is. it
cannot be the case that two different relationships hold
between two objects; furthermore, they make a full
covering of all possible topological situations, that is,
given two objects, the relationships between them must
be one of the five.
Given two geometric entities A, B in 3D and a
relationship R between them, if | «A, A, B» holds. then
«A, Ri, B» docs not hold for every RizR, and there dose
not exist a topological situation that falls outside the five
relationships.
Proof: “The topological relationship decision" tree (see
Fig. 6.) can be constructed as follow.
A°NB°=D
AnB=0 AnNB=A
touch disjoint in
F
dim(A?^B?)«
max(dim(A?)
F dim(B?)
Cross overlap
Fig. 6. The topological relationships decision tree
Every internal node in this tree represents a Boolcan
predicate, if for a certain topological situation, the
predicate evaluates to “true” then the Icft branch is
followed, otherwise the right branch is followed. This
process is repeated until a Icaf node is rcached which will
indicate to which of thc five basic relationships this
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
situation belongs. Now, two different relationships cannot
hold between two given objects. because there is only one
path to be taken in the topological relationships decision
trce. Furthermore, there can be no cases outside five
relationship. because cvery internal node has two
branches, so for every valuc of the predicate there is an
appropriate path: and every leaf node has a labcl that
corresponds to one of the five topological relationships.
The five topological relationships are expressive cnough
to represent all the topological relationships of the
dimension cxtended method. Because cach casc of the
dimension extended method can be specified by the
logical conjunction of four terms expressing conditions on
the intersection of the boundarics and the interiors of the
two objects, in general:
THOANIBIAT2(OAABIATI(ANIBIATAAAB®) — (1)
It is possible to give the equivalencies for every term TI
admissible in the dimension extended method. On the
right of each equivalence we have a logic expression Pi
making use of the five relationships between objects and
between their boundaries. Each equivalence can bc
casily tested by applying the definitions given for thc five
relationships. By substituting cach Ti with the
corresponding Pi, we obtain an expression:
PIAP2AP3AP4 (2)
expression (2) is cquivalent to expression (1). Thercfore,
the five topological relationships are able to express all
situations of thc dimension extended method.
Based on the FDS, we decided to design and implement
our own 3D GIS modeling environment on the micro-
computer. The system architecture consists of major
main components:
(1) an relation data base management system
(RDBMS);
(2) AutoCAD;
(3) an desktop mapping software ----Mapinfo
In our system, Both thematic and spatial data are
integrate within objects and stored in the same RDBMS.
Reality as capture in the data model is translated directly
to the physical RDBMS. Further, the 3D functions are
implemented within the database. Visualization and
manipulating the image for the interaction between the
computer and end-user is implemented in AutoCAD.
They are all integrated in Mapinfo and can implemented
gcographical information processing.
With the dcfinitions of formal topological relationships, a
sct of Boolcan functions including /ouch, in, cross,
overlap and disjoint can be implemented, returning
whether two objects do or do not meet the given
topological relationship. The object combinations can be
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