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4.1.1 Introducing a "surface" to the BR model 
Many geographic objects in urban space might 
be located in a relatively smaller number of 
surfaces. For example, in a terrain surface, many 
urban objects such as buildings, roads are located 
and represented by polygons. Edges in such a 
surface as a terrain surface might be able to be 
handled as if they belonged to a conventional 2D 
map data based on a planar graph. If so, polygons 
could be efficiently and reliably uncovered and the 
surfaces could also be interpolated very easily. 
The authors introduce a "2.5D surface" into the 
conventional BR model and later examine the 
possibility of the extension to a 3D surface. A 
"2.5D surface" is a surface which is represented 
by a single-valued and continuous function of co 
class defined on a 2D coordinate system, i.e. 
v=f(s,t) (figure 3). The v-axis is called a normal 
direction of a 2.5D surface. 
A planar graph in a 2.5D surface can be 
projected to a planar graph in the s-t plane along v- 
axis without changing the topological relations. 
Thus the polygons in the 2.5D surface, whether 
planar or non-planar, can be uncovered by 
applying a conventional algorithm to the planar 
graph in the s-t plane. Surface interpolation 
algorithms(e.g. TIN) for 2D data can be directly 
applied to the interpolation of their surfaces. After 
the identification of polygons and the interpolation 
of their surfaces, solids can be easily identified by 
combining 2.5D surfaces. 
T ZZ 
T 
t v =f(s 0) 
: Single-valued Fct. 
Figure 3 A 2.5D surface 
4.1.2 The data structure of the Surface 
Representation (SR) model 
In the update of a 3D spatial database, it would 
be very convenient to update topological relations 
by 2.5D surfaces respectively. To support this 
process, it is necessary to give each edge an 
attribute of which 2.5D surface contains it so that 
existing edge data can be retrieved by 2.5D 
surfaces. 
In figure 4, the basic data structure of the 
Surface Representation(SR) model 1s described 
based on the formal data structure of 3D vector 
maps [Molenaar,1990]. The data structure is the 
same as that of the conventional BR model except 
that each edge belongs to 2.5D surfaces 
respectively. In figure 4, arcs are introduced to 
avoid n to m (many to many) links between edges 
and polygons. A class denotes the class of an 
attribute of geometric features. It should be noted 
that polygons and edges can be connected with 
each other through edges even though they belong 
to different surfaces. 
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backward 
Figure 4 The basic data structure of the 
Surface Representation(SR) model 
4.1.3 Input and update of 3D spatial data with the 
Surface Representation (SR) model based 
on 2.5D surfaces 
With the SR model, the input and update of 3D 
spatial data can be done as follows (figure 5); 
(1) point data with x-y-z coordinate values and 
edge data which bound objects such as roads and 
buildings are allocated to 2.5D surfaces (figure 5, 
b),c)). 
(2) polygons, whether they are planar or non- 
planar, are identified automatically in each 2.5D 
surface, and, if necessary, attribute data can be 
given to them (figure 5,d)), 
(3) the surfaces of the polygons can be 
interpolated using the point data with (x,y,z) with 
a conventional algorithm such as a triangular 
tessellation (figure 5,d)), 
(4) after the 2.5D surfaces are automatically 
connected with each other, solids can be 
uncovered automatically in the same manner as 
with the conventional BR model (figure 5,e)) 
a) Real world 
  
j b) Point data with (x,y,z) 
and edge data 
2.5D surface c) Allocation to 
LAT 
| 
' 
2.5D surfaces 
2.5D surface 
(Side wall) 
2.5D surface 
(Terrain S 
urface) | 
  
' (Automated) 
d) Polygon 
identification and 
surface 
interpolation 
(e.g. TIN) 
    
e) Connection of 2.5D 
surfaces and solid 
identification 
  
  
  
Fig.5 Building 3D spatial database with the SR Model 
based on 2.5D surfaces