ISPRS, Vol.34, Part 2W2, "Dynamic and Multi-Dimensional GIS’’, Bangkok, May 23-25, 2001 
336 
O n (geometry, t„)= uke{0,1,...,n} P k (geometry, t k , c) 
Table 1 the steps of calculating OPH 
Instant : 
to 
ti 
12 
t'J 
tk 
Observation: 
Oo 
0, 
0, 
Oo 
Ok 
calculate PH: 
Po 
p> 
p, 
Po 
P k 
update P: 
Ho,, 
update Po 
H, r , 
Ho,, 
update P, 
update Po 
H,,3 
H.,3 
Ho,:, 
update P, 
update P, 
update Po 
Hk-,.k 
Hk-2, k 
Ho.k 
update P, 
update Po 
For arbitrary instant t n _k( k=1,2 n ), the observation 
geometry O n . k can be anti-calculated with P and H: 
O nk = ( uie{0,1 n-k} P, ) n ( D ie{0,1,...,n-k} 
(□je{n-k+1 n} Hjj )) 
3. Spatial-temporal topological model of MO 
based on consistent topological relation 
The temporal relations of two area objects have thirteen 
cases according to Allen’s event based time relation[ Allen 
J.F. 1983, ISO/TC211 1998]. 
The survival temporal period of area object is represented 
as [t s ,t e ]. Given the appearing or start instant t s of A as As, 
the disappeared or end instant teOf A as Ae, the temporal 
relation © t of area object A and B is showed as table 2. 
According to the spatial topological model of Max. J. 
Egenhofer [ Max. J. Egenhofer, 1991 ], we can define spatial 
topological relation 0 s between two MOs. Given the 
boundary of area object geometry A ( such as observation 
geometry O, present geometry P, and history geometry H) 
as 6A, interior as A , the four intersection model which 
describe the spatial topological relation of the two area 
object geometry A and B is: 
V4I = 
ÖAHÖB 
A U DÖB 
öAnB" 
A° n B° 
If the element is true in the above matrix, the value of it is 1. 
otherwise is O.The model describes eight kinds of spatial 
topological relation 0 S : 
During their evolving process, the spatial topological relation 
of the two moving object is diverse. The change of the 
spatial topological relation between two moving object at 
different instant or in different period reflects the interaction 
of two moving objects. But only the regular change of the 
spatial topological relation reflects significative interaction of 
two moving objects. We define the spatial topological 
relation that maintains consistent during their evolving as the 
significative relation, that is to say during the overlap period 
[t m » t p ] of evolving of two moving objects, the spatial 
topological relation of two moving objects maintains 
invariant. In temporal relation 0 t , there are four relations, 
before, after, meets and met-by, which have no overlap time 
period, other nine relations have overlap time period. 
We define spatial-temporal topological relation 0(© = @ s 
x ® , )of two moving object based on consistent 
topological relation during their overlap time period as 
showed in table 3. 
Given two arbitrary instant t it tj which belong to the overlap 
time period, the spatial topological relation 0 sj and © sj of 
observation geometry 0 (A) of A and observation geometry 
0(B) of B maintains invariant. 
In this model, the query of spatial temporal topological 
relation of moving objects can be represented as : A0 S B 
and A 01 B. For example: 
select A from MO 
where A during B 
and A inside B