In step (2) ‚the polygons can be identified more 
reliably than by the conventional algorithm 
[Markowsky,1980], because the reliability of this 
method cannot be affected by the coordinate errors 
of points, which often cause the failure of the 
identification of planar polygons by the 
conventional algorithm. 
Thus the complete topological relations 
between 3D geometric features Can be 
automatically established with bounding edge and 
point data obtained by digitization and/or 
surveying, except that the edge and the point data 
have to be manually allocated to 2.5D surfaces. 
4.1.4 The advantages and limitations of the SR 
model based on 2.5D surfaces. 
The advantages of the SR Model are 
summarized as follows. 
(1)Higher efficiency of the input and update of 3D 
spatial data . 
A 3D spatial database can be built and updated 
very efficiently only by providing edge data and 
point data with x-y-z coordinate values as 2.5D 
surfaces. 
(2)Ease of representation of urban space 
With the SR model, urban space can be 
represented easily by geometric features such as 
edges, polygons and solids in the same manner as 
with the BR model. 
(3)Integrated use of 2.5D map data in a 3D spatial 
database. 
2.5D map data can be stored and simultaneously 
utilized in a 3D spatial database with the SR model 
because the 2.5D map data can be stored as a 
single 2.5D surface. 
The limitation of the SR model based on 2.5D 
surfaces comes from both the manual allocation of 
edges and points into each 2.5D surfaces and the 
influence of coordinate errors on the connection of 
2.5D surfaces. For example, a building and a 
terrain surface with a step must be divided to 2.5D 
surfaces unnaturally (figure 6). Although an 
example of urban space modelling shows that both 
are not severe, the authors examine the possibility 
of extending 2.5D surfaces to 3D surfaces in the 
next section. 
  
Figure 6 Decomposition to 2.5D surfaces 
4.2 A possibility of extension of 2.5D surfaces to 
3D surfaces 
1) Constraint conditions on 3D surfaces to ensure 
semi-automatic polygon identification 
260 
  
Representation by 
2.5D surfaces 
Representation by 
3D surfaces 
Figure 7 An advantage of 3D surfaces in 
represeting 3D spatial objects 
A 3D surface is a surface which can be 
embeded into a plane or a sphere with holes. It 
does not intersect with itself and so an edge in a 
3D surface bounds with no more than three 
polygons. By 3D surfaces, 3D spatial objects can 
be represented more naturally and easily (figure 
7). However, several limitations must be given to 
3D surfaces to ensure that the polygon 
identification and the surface interpolation can be 
done semi-automatically. The limitations can be 
summarized as follows. 
(1)Constraint conditions on the number of edges 
at one point: 
i)The number of edges which start or end at 
one point should be less than three or, 
iif the number of edges is more than three, the 
"order of edges over a surface " must 
coincides with the "order through the 
projection". 
The "order over a surface" is obtained by 
tracing the edges over a surface formed by 
them (figure 8). The "order through the 
projection" is obtained by ordering projected 
edges on a projection plane which is a rough 
approximation of a surface formed by the 
edges, in a counterclockwise or clockwise 
manner as shown in figure 9. Figure 10 is also 
an example where both orders of edges agree 
with each other, while figure 11 shows an 
example of the disagreement. 
Figure 8 Ordering edges over a surface 
clockwise or 
A 
I 
I 
C ; > counterclockwise 
: ordering 
  
Figure 9 Ordering edges through 
the projection to a plane