ISPRS, Vol.34, Part 2W2, "Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
Definition 3: The geometry part of MO which formed at t| and 
continues existing until tj, disappeared after tj, that is to say 
the part existed during period [ti, tj], defined as H, we 
named it history geometry: 
Hi. j( Geometry, tj, tj) e object2 
Example 3: the geometry which existed during period[t 0 , ti], 
and became history after instant ti is H 01 (geometry, t 0 , ti), as 
showed in Figure 1(d). 
So, a moving object composes three parts: 
MO={ UOi ( Geometry, tj ), uPi( Geometry, t| , 
c), u Hi. j( Geometry, ti, tj) } 
Here u O, ( Geometry, ti ) is the union of geometry 
observation snapshot at a series of instants, u Pi( Geometry , 
ti , c ) is the union of geometry formed at previous instants 
and continue existing at current instant. U Hij( Geometry, t,, 
tj) is the union of geometry of history during periods. The 
three parts of MO are represented as MO O, MO.P and 
MO.H. In following calculating, we simply replaced as O, P 
and H. 
2. 3 Calculating and updating of OPH 
(3) Observation occurred at instant t k , current instant is c= 
tk+5, and c*= tk-i+5, the observation geometry of area object 
is Ok ( Geometry , t k ), to calculate P and H: 
(a) The new part that appears at instant t k , but did not 
appear at instant t 0 , ti, , tn and continuously exist at 
current instant, represented as P k (Geometry, t k , c). which is 
the geometry point set difference of O k (Geometry , t k )and P 0 , 
Pi Pk-,: 
P k (Geometry, t k , c) = O k (Geometry , t K ) -Di 
e{o,i k-D P i (Geometry, t, c*) 
(b) The history geometries that appeared at instant t 0 , 
t,, , tk-i but disappeared at instant t K , represented as H 0 .k, 
Hi.k, Hk-i.k, the calculation of them as: 
H k .i. k (Geometry, tu, tk) 
H k -2, k (Geometry, t k . 2 , tk) 
H 0 , k (Geometry, to, t k ) 
kx 1 
Pm (Geometry, t k -i, c*) 
P k . 2 (Geometry, t k . 2 , c*) 
P 0 (Geometry, t 0 , c*) 
OkGeometry , t k ) 
O k Geometry , t k ) 
-KX1 
(1) At first observation instant to, the observation geometry of 
MO served as original condition, that is to say current instant 
c= to+6, we have P 0 = O 0 , because there is only one 
observation instant, there is no new formed geometry and 
no disappeared as history geometry. At instant t 0 MO 
represented as: 
MO={ O 0 ( Geometry, to), P 0 ( Geometry, to, c ), <D } 
Here: P 0 ( Geometry , to, c ) = O 0 ( Geometry, to). 
(2) Next observation occurred at instant ti, current instant c= 
to+6 is updated with ti+5, and c*= to+5, using the observation 
geometry Oi at instant ti, to calculate P and H: 
(a) The geometry Pi ( Geometry, ti , c) that did not 
appear at instant t 0 but appeared at instant ti, and 
continuously existing at current instant. That is to say Pi is 
the part of observation Oi which did not exist at instant to,, 
calculating of Pi is the geometry point set difference of Oi 
( Geometry , ti ) and P 0 ( Geometry, t 0 , c*): 
Pi ( Geometry, ti , c ) = Oi ( Geometry , 
ti)- P 0 (Geometry, t 0 , c*). 
(b) The history part of geometry that appeared at instant 
to and disappeared at instant ti, according to definition 3, 
which represented as H 0 .i( Geometry, to , ti). Calculating of 
it is the geometry point set difference of P 0 (Geometry, t 0 , 
c*)and OkGeometry , ti): 
H 0 ,i( Geometry, t 0 , ti ) = P 0 (Geometry, to , c*)- 
Oi(Geometry, ti) 
(c) The part of geometry that appeared at instant to and 
continuously exist at observation instant t 1t which actually is 
to update P 0 , Calculating is the geometry point set 
intersection of P 0 (Geometry, t 0 , c*)and OkGeometry, b): 
Po(Geometry, t 0 , c) = P 0 (Geometry, t 0 , c*) n 
OkGeometry , ti) 
Then at instant ti the moving object MO is represented as: 
MO={ uie{0,1}Oj( Geometry, t ),uie{0,1}Pi( Geometry , 
t, c), Ho.k Geometry, t 0 , ti )} 
Ok (Geometry , t k ) 
lTxi 
(c) The geometry part that appeared at instants to, 
ti, , tk-i, and continuously exist at current instant, which 
is actually to update P 0 , Pi, Pn- The calculation of 
them: 
Pk-kGeometry, tk-i, c) 
s' -\ 
Pu (Geometry,tu, c*) 
P k . 2 (Geometry, t k . 2 , c) 
Pk. 2 (Geometry,t k - 2 , c*) 
(Geometry, t 0 , 
c) 
kxi 
P 0 (Geometry,to, c*) 
kX 1 
Ok (Geometry, t k ) 
n 
Ok (Geometry, t k ) 
Ok (Geometry, t k ) 
kx 1 
At instant t kl after calculating and updating OPH, MO is 
represented as : 
MO = { Ule{0,1,...,k}Oi( Geometry, ti), 
Uie{0,1,...,k} Pi( Geometry, ti, c), 
uie{0,1 k-1}(U je{i+1,i+21,...,k} Hj,j( Geometry, 
ti.tj)) } 
The steps of calculating and updating P and H at each instant 
are showed as in table 1. 
At instant tk, the number of geometry set operation is 2K-1. 
2. 4 Anti-calculation of O with P and H 
The present geometry P and history geometry H of MO are 
calculated by observation geometry O of MO , so P and H have 
composite relationship with O. 
If current instant is t n , the observation geometry O n can be 
anti-calculated with P: 
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