International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
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3.2.2 Equality and inequality constraints: Each class of 
object representation (see Table 1) has its own constraints 
which will be derived in the following. 
Horizontal plane: Heights of objects which represent a 
horizontal plane must be identical. This means, that points P; 
with a height Z; and planimetric coordinates X; Y; situated inside 
the object boundary (see Figure 4a, black points) must all have 
the same value Z,, which has to be estimated in the 
optimization process. These height values lead to the following 
observation equation: 
0$, 2 ZZ,» -Z, (3) 
Additionally, the heights of the bounding polygon points of the 
topographic objects must be identical to the height of the 
horizontal plane. The height difference between the unknown 
object height and the calculated height is used to formulate an 
additional pseudo observation (see Figure 4a, dark grey points): 
^ 
0 t De = Z up r7 Z, (x 
Y 4 
m^^m u? 
A Z, ) (4) 
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$ e e * » e 
e * 
9 9 e e. € » e € 
* * 
a. e € ® e € 
2 e 
noe e € € 4 
Figure 4: Equality and inequality constraints of an object “lake” 
The neighbouring terrain of the horizontal plane is considered 
using the basic observations (1) and (2) (see section 3.2.1). If 
the object represents a lake it is necessary to use a further 
constraint which represents the relation between the lake in 
terms of a horizontal plane and the bank of the lake whose 
unknown height values 2 have to be higher than the height 
level of the lake: 
0 > Z zZ (5) 
It is set up for all points marked in black in Figure 4b. 
Tilted planes: The objects treated in this paper which can be 
composed of serveral tilted planes are elongated objects. In 
longitudinal direction these objects are not allowed to exceed a 
predefined maximum slope value sy 
Z,-2 
za (6) 
no 
The example in Figure 5 shows a road which is modelled by 
lines and then buffered using the attribute “road width” of the 
GIS data base. Here, Z, and Ze are the unknown height values 
of successive points P, and P, in driving direction of the road 
(Figure 5a). D,, is the horizontal distance between these points. 
a) b) 
P. P: | / p. 
Figure 5: Equation and inequation constraints of an object 
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road 
In addition, the difference between two successive slope values 
which is comparable to the vertical curvature of the object is 
restricted to the maximum value ds,,,, (Figure 5a): 
> 
u$, D r£, 
d$ ue 7 E : =r (7) 
Ho op 
In case of a road, the points P,, P, and P, are successive points 
of the middle axis of the object. 
Assuming a horizontal road profile in the direction 
perpendicular to the middle axis the height values of 
corresponding points must be identical: 
042, — Z, -Z, (8) 
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The values Z, and Z, represent point heights of the centre 
axis and the left or the centre axis and the right side of the 
buffered object (Figure 5b). These constraints are introduced for 
all cross sections whose centre point results from the 
intersection between the DTM TIN and the object centre axis 
(Steiner points). Those cross sections whose centre points are 
original points of the object middle axis are not used to form 
this kind of constraint because in the original points the road 
may show a change in horizontal direction and slope (Figure 
5b, profile p»). Consequently the cross section is not horizontal. 
Finally, the points of any two neighbouring cross sections and 
the points in between have to represent a plane: 
03, sd da X, c 0,Y, -Z, (9) 
In Figure 5b the points of the neighbouring profiles p; and p, as 
well as the red point in between represent a point P, of equation 
(9). These points have to represent a plane with the unknown 
coefficients à,,à,,@, . X,, Y, are the planimetric coordinates of 
point P,, 2. is the height of P, which has to be estimated. A 
special case is the treatment of the points of a cross section 
involving an original object point of the 2D road centre axis. As 
an example let's consider the profile p; of Figure 5b. Equation 
(9) is set up twice, once for the horizontal profile p; and the 
centre point of profile p; (and any point in between), and again 
for the horizontal profile p; and the centre point of profile p» 
(and any point in between). After the optimization process the 
intersection line of the two neighbouring planes can be 
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