we can define the Hausdorff directions and the dynamic 9- 
intersections for directional relations as follows: 
q((K,K.)sinfía: K, CK, 9€ R(atz)K,cK, e en [9] 
Q(K, Ks KK) 
[K,GR(a)^K? [K R(eyn^6éK, [K@R(a,)]°NK; 
RL (2) AKOR@)INK; AK OR(@)NK, AK OR )NK, 
[K,®R(a,)] "kK; [K®R(«)] NK, [K®R(«)] NK, 
[10] 
KeCpK,OR(«,) Kr K,R(a,) KK, ®R(et,)] 
RY, (4 )H &K,nLK, 6R(a ))] £X, AL K,R(«,)] ÉK,NK,®R(a,)T 
K;NK,@R(a,)]° K, NÉ[K,@R(a,)] K?[K,®R(a,T 
[11] 
where the K®R(«,) means relevant dilation by the closed 
angular bearing R with angle «, , and the Re (o) means the 
dynamic 9-intersection for directional relations with parameter 
a, from K, to K, . Based on equations [10] and [11], we also can 
derive dynamic topological relations by using the different 
parameter «, . 
Similarly, according to the derived dynamic topological 
relations by the dynamic 9-intersections of equations [10] and 
[11] with different parameter «,, we also can simply get the 
Hausdorff direction p(K,,K,) (or o( K,, K.) )between two closed 
sets K, and K, by calculating the minimum and maximum 
dilated angles based on equation [9]: 
9 (KK, )2max(min(a,), min(a ,£z)5 
p(K, K )=p(K, KR, yen 
100][000] [111][011] [oor] [oor][111][1*1 
95,5 (6,)7] 010 1010 |001 001 |, OTI S | ITT} 01 v *1L E. [12] 
| 001 ||001 | [001||001| [001] |001||001||001 
100 |[000] [100][000][000| [010][110|[1+0 
Rean(«,)= 010 || 010 |, |100 || 100 | 010 |, |010 {100 || #10 | 
; 001 |[001| [111 |[111 |[111} [111 |[111 }[111 
TT M 
k when 
equal contains convers 
where "*" means either empty (0) or non-empty (1). The 
binary directional relations derived by [12] are also suitable for 
different types of spatial objects and their combining types. 
Some examples are shown in Fig.5. 
3.3. Order Relations between Sets 
Formal methods for the description of order spatial relations 
can be based on mathematical theories of partially-ordered sets 
and lattices. The use of greatest lower bounds and least upper 
bounds for describing order spatial relations shows that we 
need a lattice in order to find an answer in all possible cases. 
Since not every partially-ordered set 1s a lattice, it is, however, 
always possible to add elements to a partially-ordered set to 
create a lattice. The process of normal completion specifies how 
to find the smallest numbers of elements necessary to add to a 
partially-ordered set to create a lattice, ie. to build the 
minimal containing lattice of a partially-ordered set. The 
general descriptions of order relations can be found in Kainz et. 
al. [1993]. In this paper, we emphasize to study the problems of 
the detail classifications of order relations and their integration 
with other kind of spatial relations. 
Based on metric and topological relations, order spatial relation 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
  
  
  
  
  
  
  
  
  
(a) _P(AB)=& (e) 4 
7 € N v Ep, 5x iid N 
{ è À I $234 2 B X 
i A Bii AC IB } A $5 A ap } 
NL L4 p. \ ' 
E re @(A, B)=as elt A 
(b) 
ES 
7 s. 
{ 5i 
S ei 
Doi J pain, (54 MA PEU à 
po x p Ä ~ L0 
7 B GC \ B t0 B X B Ó 
: A 
1 ; i 
; Q } 
^ A ; A X Aa=21 A 
BG : z = 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Fig.5. To obtain the Hausdorff distance and directions between two 
planar sets 4 and B. 
can be classified to the three kinds, i.e. distance orders (such as 
the areas of regions or the lengths of lines), directional orders 
(such one spatial object at left/right or in front/behind to 
another one), and topological orders (such as one spatial object 
is inside another or its inverse operations whether a spatial 
object contains another one). Let N be the greatest lower bound 
and U be the least upper bound of the completion lattices, Fig.6 
shows. some examples of lattice resulting from the normal 
completion of order spatial relations measured by distances, 
directions and topological covers. 
4. MODEL EXTENSIONS 
4.1. Metric Relations between Subsets 
According to the fuzzy set theory [Zadeh, 1965], the concept of 
distances or directions between subsets are fuzzy, since the 
spatial objects may contain many subsets, the distances or 
  
(a). distance orders 
  
     
  
B1 
  
  
ALB 
Jars az 
  
    
70 rio2 5 
(b). directional orders 
  
(c). topological orders (v) 
B 
HG (A) (B) 
A LN 419 
«>» 
  
  
  
Fig.6. The classification of order spatial relations. 
102 
    
  
   
    
   
  
  
  
  
  
   
  
   
  
  
  
   
   
   
   
   
   
   
    
  
    
  
    
   
   
   
    
   
   
  
   
    
   
  
  
  
   
   
   
  
  
   
   
  
   
   
    
directions be 
just by a sing 
as the coveri 
numbers and 
R^, we can 
case with th 
the changes 
and line len 
estimate the 
can quantita 
For reasons 
closed sub-r 
for estimati 
follows: 
o, (195: 
o, 0-1 
where A(*) 
with the rt 
separately, 
the parame 
called the s: 
An exampk 
is shown in 
4.2. Distan 
In a space 
between ob 
the path crc 
does not ci 
between a 
constrained 
possible co 
intersect th 
called the " 
distance d, 
p, and p. 
linking p, 
the descri 
geodesic € 
distance fi 
distance d, 
distance b 
constraine 
distance b: 
1991; Lant 
The geode 
space X ar 
empty con 
compact s 
quantity: 
pe (KK,