(v)The surface interpolation of polygons in each 
3D surface: 
If a polygon is convex, the surface is 
interpolated by iterating the generation of 
triangular planes by connecting neighboring 
edges. When a polygon is not convex, i.e. a 
polygon is concave and/or other polygons and 
points are contained in the surface of an object 
polygon, the polygon is divided into concave 
polygons by adding edges to connect points 
where the intersecting angle of edges is over 
180 deg. 
Since there are possibilities that some data may 
violate the constraint conditions, the normal 
vectors at points must be displayed to ease the 
user's check of the result of edge ordering. With 
this procedure of uncovering polygons in a 3D 
surface, the SR model based on a 3D surface can 
be implemented. 
5 INTERPOLATION OF ELEVATIONS IN A 
DIGITAL URBAN SPACE MODEL 
5.1 Introduction 
A large amount of elevation points are 
necessary to represent 3D spatial object with a 
DUSM. Especially the representation of terrain 
surfaces requires many reliable elevation points. 
This is not only because terrain surfaces have 
complicated shapes but also because the elevation 
of other spatial objects such as underground 
structures have to be determined based on the 
elvations of terrain surfaces. 
However it is no easy task to assign elevation 
data to many points manually. For example, it is 
very labor-demanding to obtain elevation data 
from conventional maps in urban areas because 
contour lines are usually cut in pieces due to 
buildings and other man-made features. With 
aerial surveying techniques, it is not so easy to 
obtain enough number of elevation points due to 
occlusions. Only roads and the roofs of buildings 
are exceptionally easy place for 3D measurement. 
A method of elevation interpolation in urban areas 
is indispensable to reduce the requirement of 
elevation data and to give a sound basis of 
elevation to a DUSM. 
5.2 A method of elevation interpolation 
Existing surface interpolation methods usually 
assume that terrain surfaces are smooth although 
the discontinuities of slopes and elevations are 
often the case in urban areas. To make larger-scale 
representations of terrain surfaces and related 
spatial objects in urban areas, the following 
geometric conditions must be considered, which 
characterize terrain surfaces in urban areas (figure 
15). 
Break lines: The steepness of slopes shows 
discontinuities on a break line. 
Break lines are often to be seen in 
the boundaries of man-made objects 
such as roads and levees. 
Step lines: Elevation shows a abrupt change 
(like steps) on a step lines. Retaining 
262 
Horizontal plane Break line 
   
Step line 
Figure 15 Examples of geometric constraint conditions 
in elevation interpolation 
walls and the side walls of buildings 
are generated by step lines. 
Horizontal planes: Every points in a horizontal 
plane has the same elevation value. 
Floors of buildings are the 
examples. 
Under these geometric conditions, surfaces are 
represented by TIN to easily integrate the 
interpolated surfaces with a DUSM. At places 
where these geometric conditions do not hold, 
elevations are interpolated under the assumption 
that a terrain surface is smooth. Smooth terrain 
surfaces are obtained to maximize the sum of the 
square of inner-products of unit normal vectors of 
neighboring triangular planes. Moreover, some 
lines such as road boundaries sometimes have to 
be interpolated smoothly. The "smoothness" of 
lines is evaluated in terms of the sum of the 
squares of vertical changes of unit vectors along 
the lines. Thus elevations are interpolated so as to 
maximize the smoothness of terrain surfaces and 
lines under the above geometric constraint 
conditions. 
6 AN EXAMPLE OF URBAN SPACE 
MODELLING 
An example model based on a 2.5D surface has 
been made of Nishi-Shinjuku, which is one of the 
busiest business and commercial districts in the 
Tokyo Metropolitan Area (figure 16). The size of 
the example area is about 1.5km by 2.0km. 
Figure 17 is a 2.5D surface representing the 
terrain surface and the ground floors of buildings. 
Polygons are uncovered in the edges and given 
attribute data of categories of floor-uses . Figure 
18 shows the result of the surface interpolation. 
The total number of triangular polygons 
representing the terrain surface is more than two 
thousand even though the example area is not 
large. With the conventional BR model, a human 
operator would be required to generate edges 
bounding many triangular polygons by connecting 
an enormous number of elevation points. 
In this example, several important 2.5D 
surfaces such as those of the terrain surface, the 
first and the second basement floor and the second 
floor etc. were input by the digitization of existing 
maps. But many of the other surfaces such as 
those for the other floors of buildings could be 
generated with a "copy" command using some 
additional data such as elevation data of the floors. 
Pt 
1x2