Full text: XVIIIth Congress (Part B3)

three geometric types: nodes, edges and faces 
represented by the symbols n, e and f 
- three geometric object types: point objects, line 
objects, area objects. 
The further developments will only use line- and area 
objects which will be represented by the symbol O, and 
O, . Instances of these entity types will be indicated by 
suffixes. 
The reader will recognise that the formalization explained 
in this paper is to a large extent isomorphic with topologic 
data structures defined for GIS such as ATKIS/DLM, DGF, 
TIGER and DIME etc., see (Hesse 1992, Walter 1994, 
Marx 1990, USBUREAU 1990). This formalization will be 
based on the FDS described in (Molenaar 1989). 
Relationships Between Nodes, Edges and Faces 
The following relationships can be defined between the 
geometric elements of a planar graph: 
> Edge e, has node n, as the begin node 
^ Beginle, n] — 1 otherwise = 0 
- Edge e; has node n, as the end node 
- Endle, n] = 1 otherwise = 0 
We will consider edges as straight line segments. Each 
edge will always have one face at its left hand side and 
on at its right hand side. These relationships will be 
expressed by the following functions: 
- Edge e, has face f, at its left-hand side 
—1iele,f.lo-7 
For any f, z f, we get then Le/e,, f,] = O 
- Edge e; has face f, at its right-hand side 
- Rile,, f,] - 1 
and again for f, z& f, we get then Rile,, f,] = O 
If an edge e, has face f, at the right hand side and face 
f, at the other side then these faces are adjacent at this 
edge, which will be expressed by the function 
ADJACENTIY,, f,le;] = 1 (and = O otherwise) 
the fact that there is some edge where the faces are 
adjacent can then be expressed by 
ADJACENTIf,, f,] = 1 (and = O otherwise) 
Line Objects 
The geometry of a simple line object is represented by a 
chain of edges as in figure 1a. The fact that an edge e, 
is part of the object can be established by the function 
Part,, [e,, O,]. The notation Part, [ ] means that an entity 
with spatial dimension u is a part of an entity with 
dimension v. If the edge is part of the object then 
Part,,[ | = 1, else it has a value = O. 
À line object will have a begin node 
n, = BEG(O,) and an end node 
n, = END(O,). These can be found through the edges of 
O,, the direction of the object can then be specified by 
DINO Mn} 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
    
a. relationship between edge and line object 
   
  
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b. relationship between edge, face and area object 
area object a 
    
  
FACE right 
  
area object b 
fig. 1: Relationships between edges and objects. 
Area Objects 
The geometry of an simple area object is represented by 
one or more adjacent faces as in figure 1b. If a face f, is 
part of an area object O, this will be represented by 
Part, [f,, O,] — 1. The set of all objects from which a 
face fis a part is OA(f) = { O | Part,,[ £ O] = 1} 
This function relates two two-dimensional entities. If there 
are overlapping area objects then each face might be part 
of several objects, but each object will also consist of one 
or more faces. Therefore this is a many-to-many relation- 
ship. Overlapping objects can be found through their com- 
mon faces. 
Now it is possible to check whether edge e, is related 
through face f, to an area object O,. There is at most one 
face for which both Ze/e;, f,] = 7 and Part,,[f,, O,] =1. 
If such a face exists then the function relating the edge 
to the object will get the value — 1, in all other cases it 
will be — O. Hence if edge e, has area object O, at its left- 
hand side then Le/e, O,] = 7 else = 0. Similarly if edge 
e, has area object O, at its right-hand side then A//e;, O, 
] 21 else « O. 
The combination of these two functions gives for edge 
e; : 
Ble,, O,] = Lele;, O,] + Rife,, O,] 
If an edge e, is part of the boundary of O, then only one 
of the functions Ri and Le is equal to 1 but not both, so 
for such an edge we find Ble;, O,] = 1. If e; has O, both 
at its left-hand side and at its right-hand side then 
Ble;, O,] = 2, in that case it is running through O,. If 
Ble;, O,] = O0 there is no direct relationship between e, 
and O,. 
Adjacent Area Objects 
When an edge has an object O, at its left hand side and 
not at its right hand side and object O, at its right hand 
side and not at its left hand side then these objects are 
adjacent at this edge. If the objects overlap not at all, i.e. 
if they have no common faces and they are adjacent at 
least one edge, then they are adjacent which is expressed 
by the function ADJACENT[O,, O,] = 1 (and = O 
  
  
    
   
    
     
  
     
    
     
     
    
  
    
     
   
  
  
  
  
    
   
  
   
  
  
    
     
    
   
    
    
  
  
   
   
    
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