relationships between two geometric objects for 3D GIS. 
Twelve conventions for FDS have been formulated. Such 
as the arc are geometrically represented by straight line 
segments, but arcs are not allowed to form loop; Two arc 
may not intersect; Faces arc planar and should not 
intersect, etc. These support the unambiguous mapping of 
the terrain situation to a FDS database. 
3D FDS is an attempt in 3D modeling suitcd for gco- 
information in many aspects. However, it was not 
designed to also handle complex objects and 
interpolation processes efficiently and creating the data 
model from raw input data and ensuring data 
consistency is still a big challenge. 
4. THE FRAMEWORK OF TOPOLOGICAL 
RELATIONSHIP AND 3D GIS 
People have done many works in description. of 
topological relationships (Guting, 1988; Pullar, 1988; 
Wagner,1988; Egenhofer and Franzosa, 1991) A 
drawback of these methods is that they distinguish only 
between empty or non-empty intersections of the 
boundaries and interiors of two geornetric objects, it is 
not carried out in all its consequences. For example, no 
distinction can be made in Fig. 3. and Fig. 4. because 
both relationships have same description. 
  
  
   
Fig. 3 Fig. 4 
Clementini et al. adopt the calculus-based method to 
group together the relationships into a few more general 
topological relationships: touch, in, cross, overlap and 
disjoint. This method is a good formal description of 
topological spatial relationships, but it is in analogy 
with the 2D situation. 
In this paper, we take into account the dimension of the 
result of the intersection ( dimension extended method) 
in 3D space, then we give the exact semantics to the five 
basic topological relationships and prove that five 
topological relationships are mutually exclusive. Finally, 
we give the framework of 3D GIS modeling environment. 
Formal definitions of geometric objects and relationships 
are based on the point-set approach. The topological 
space 1s IR The boundary and thc interior of 
geometric objects are used in the dimension extended 
method for describing the topological relationships. The 
boundary of a Point Object to be always empty; the 
boundary of a Line Object is an empty set in the case of a 
circular line while otherwise is the set of the scparate 
end-points; the boundary of an Surface Object is a 
circular line consisting of all the accumulation points of 
the Surface: the boundary of a Body Object is a circular 
arca consisting of all the accumulation point of the Body. 
We consider a function “dim”. which returns the 
dimension of a point-sct. In casc the point-sct consists of 
multiple parts, then the highest dimension is returned. In 
the following definition, S is a general point-sct. which 
may consist of scveral disconnected parts: 
Oifs © 
| 0 1f S contains at least a point and no lines , arcas or bodies 
dim(S) 11 if S contains at least a line and no areas or bodies 
2 1f S contains at least a area and no bodies 
3 1fS contains at least a body 
The boundary of a Object A is denote by dA. The interior 
of a Object A is denote by A?. It is defined as A°=A- 
OA. Suppose the intersections of thc boundaries and 
interiors of the Object A and B are represented by thc 
four sets: 
Sl-0^Anc0B; S220AnB*, S35 A?n0B; |. S45 A??? 
In the dimension extended method we take into account 
the dimension of the intersection, so Si (i=1.2,3.4) can 
be O,0,1,... dim(Si). If Si ( 171, 2, 3. 4) have Ni 
different cases, the topological relationships between the 
object A and B have N! x N2 x N3 x N4 = N possible 
case. Apart from some impossible case, all results in a 
total of real cases. 
Suppose the notation <A, R, B> mcans that the objects 
A and B are involved in the relationship R; this triplet is 
called a fact. fact can be combined through the and(^), 
or(v) Boolean operators. For the 3D case, the dcfinition 
of five relationships are the following: 
Definition 1. The fouch relationship: 
«A, tuoth, B» > (A°NB°=O)A(ANB=#O) 
Definition 2. The in. relationship: 
«A, in, B» €» (A?^B?zO)A(A^OB-A) 
Definition 3. The cross — relationship: 
«A,cross,B» €» dim(A?^3B?)«(max (dim(A?).dim(B?)) 
ANANB#A) A(ANnB#B) 
Definition 4. The overlap relationship: 
XA,overlap, B» €» (dim (A?^3B?)-dim (A?)-dim (B?)) 
A(ANB#A) A(AnB#B) 
Definition 5. The disjoint relationship: 
<A, disjoint, B> © AnB=0 
Some examples of topological spatial relationships 
between Line Object with Body Object can be seen in Fig. 5. 
276 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
    
   
   
   
    
   
  
  
  
    
   
     
   
   
   
    
   
      
    
   
    
    
    
  
  
   
    
   
    
   
    
    
   
   
   
   
  
    
     
   
    
   
   
    
(a 
  
The fiv 
cannot 
betwee 
coverir 
given t 
be one 
Given 
relatioi 
«A, Ri 
not exi 
relatioi 
Proof: 
Fig. 6. 
—Q 
  
Fi 
Every 
predic 
predic 
follow 
proce: 
indica