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Line- and Area Objects 
Several important relationships between a line object O, 
and an area object O, can be found by checking for each 
edge that is part of the line object how it is related to the 
area object. This will be expressed by the functions 
Le[O,, O,| e;] = MIN(Lele;, O,], Part,,[e,;, O,1) 
Ri[0,, O, | e;J = MIN(Rile,, O,1, Part,, le;, O,1) 
For the relationship between a line object O, and an area 
object O, we can write 
BIO, 5:0, les: a LelO Oy edt RIO, Oye 
If this function has the value — 2 then the line object runs 
through the area object at edge e,if the value = 1 then 
it is at the border and if it is = O then there is no 
relationship. The relationship between the two objects 
might be different at different edges. 
A Hydrologic Example 
For modelling hydrological systems three types of 
elementary objects will be defined according to (Martinez 
Casasnovas 1994), these are the water course lines, the 
drainage elements and their catchments, see figure 2. The 
drainage elements are gullies, each element has a catch- 
Catchment Area ~~ | * 
/ dai Y 7 L-—7r- Water Course Line 
We 4 Drainage El. 
Starting node, —- | rer Area 
From node or | baie 2 
Inlet point-- POOR : 
i 
ios a End node, 
LT To node or 
Outlet point 
fig. 2:Elementary objects in a drainage system. 
ment area from which it receives overland flow of water. 
Each element also receives water from upstream elements 
(if there are any) and it empties into a downstream 
element. The water flow through each element is repre- 
sented by a water course line. 
The relationship between these objects is one to one in 
the sense that each drainage element D; contains exactly 
one water course line W, and is embedded in exactly one 
subcatchment area C,. A subcatchment area may be 
dissected by its drainage element, as can be seen in figure 
2, but it is still considered as one subcatchment. The 
topologic relationships between these objects can be 
expressed by functions of section 2: 
for water course line W, and drainage element D, is: 
(ve, | Part,,[e,, W,] » 115 BIW, Dj|]e,] - 2 
this will be written shortly as BIW,, D;] — 2 
ifj # i then B[W;, D;] = O 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
This means that W, runs through D; so that it has D; at both 
sides and it is not related to any other drainage element. 
This is a topologic restriction due to a semantic constraint 
valid in the context of this hydrologic model. Another 
semantic constraint is 
for drainage element D, and catchment C, is 
ADJACENTID,, C;] — 1 
if j # i then ADJACENTID,, C;] = 0 
so that D,is only adjacent to C, and to no other catchment. 
Each drainage element is also connected to a downstream 
element and, depending on its position in the network, to 
one or more upstream elements. The relationship between 
the drainage elements can also be found through the 
watercourse elements. These should be directed according 
to the direction of the water flow, for each W, we can find 
the upstream element W, through the rule END(W, ) = 
BEG(W,). This relation between these water course lines 
will be expressed by Upstr[W,, W,] = 1, this function will 
have the value = O otherwise. 
Due to the 1 to 1 relationships between W, D and C the 
upstream relationship can be transferred as follows 
Upstr[W,, W,] = UpstriD,, D,] — Upstr[C,, C;] 
so that the order relationships between the water course 
lines can be translated into order relationships between 
the areas in which they are contained. We will assume 
here that the stream network structure is defined so that 
for each W; with a Strahler number > 1 there are two or 
more upstream water lines W;, but for each W, there is 
only one downstream water line W,. 
2.2. Object Classes and Class Hierarchies 
Terrain objects refer to features that appear on the surface 
of the earth and are interpreted in a systems environment 
with a thematic and geometric description. In most applica- 
tions the terrain objects will be grouped in several distinct 
classes and a list of attributes will be connected to each 
class. Let Ci be a class, and let the list of its attributes 
be L/ST( Ci) — (A, As....., AJ then 
LIST( Ci ) # LIST( Cj ) for i # j 
i.e. these attribute lists will be different for different classes. 
Terrain objects inherit the attribute structure from their 
class, i.e. each object has a list containing a value for each 
class attribute, thus for member e of class C: 
LIST(e)] i:(845-85..-, 45) 
where: a, = A,le) is value of A, for object e 
e &€ 
A, E-LISTIC) 
When two or more classes have attributes in common, 
then a superclass can be defined with a list containing these 
common attributes as "superclass-attributes" (Molenaar 
1993). The original classes are subordinated to these super 
classes, for example, the class 'forest' is a superclass 
containing subclasses such as "deciduous", "evergreen", 
and "mixed forest". The terrain objects are then assigned