Duration Mutation 
The Duration of Arc al 
  
  
1 
I 
1 
The Duration of Arc all : 
I 
1 
  
The Duration of Arc a2 
1 
1 
! 
SS ENS NN 
1 
1 
S NS 
  
The Duration of Arc a5 
em om 
The Duration of Arc a4 
The Duration of Arc a41 
The Duration of Arc a42 
A4 
5 -— 
(CE ALL ML LA LL EE 
  
  
  
The Duration of Arc a3 
  
  
The Thematic Time 
Durations of Polygon A 
! 
I 
I 
I 
| 
I 
i 
1 
I 
I 
i 
1 - 
( 
! 
i 
! 
' 
I 
! 
i 
! 
The Durations of Polygon À 
TO 
   
Figure 3.2 The Time Topology of Polygon A 
4. THE REPRESENTATION OF OBJECT STATES 
WITH HISTORICAL RELATIONS 
The representation of object states here is based on the 
tuple-based world time stamping approach. The state and 
time topologies provide the fundamental for storing geo- 
graphic information. The states of an object may be 
represented as the contents of the relations, while the time 
durations of the object may be represented as the tuples’ 
intervals. 
We keep the topological data, location data, and attribute 
data separately in different relations. Therefore, to record 
one kind of data does not duplicate the others, and to 
retrieve one kind of data does not require movement 
through the others. Furthermore, different topological rela- 
tionships, such as polygon topology, arc topology, or node 
topology, may be independently stored, as are the attribute 
data. Consequently, to record or retrieve one kind of topo- 
logical relationship does not require reference to the others. 
We assume that each arc has only one state. The term 
al[Ti, NOW) is used to show that arc al was born at time 
Ti, and is now still alive. À tuple which has time interval 
[Ti, Now) is called an active tuple. If, finally, arc al died 
at time Tj, then only its time duration needs to be amended 
to al[Ti, Tj). The corresponding tuple is changed into a 
historical tuple. The historical tuples can only be 
retrieved; they cannot be changed. The active tuples can be 
retrieved, and also be amended into historical tuples. 
Therefore, we represent the evolution of a geographic 
object over time by recording its changing information. For 
example, the states of polygon A changes over time can be 
represented in database as Figure 4.1. 
159 
Since we use a set of relations to represent the states of a 
single data layer. If every tuple of these relations embeds 
with a world time interval, a large amount of time interval 
duplication will exist. In order to reduce this time interval 
redundancy, we classify the set of relations for a data layer 
into two categories, and embed the world time only on the 
tuples of the relations in one categary. Then use join to 
propagate the time attributes to the relations of another 
categary. For example, in Figure 4.2, although the relation 
ARC-ATTRIBUTE does not have time attributes, it con- 
tains the historical information, such as arc a2 which had 
changed into a'2 at time T1. The duration of arc a2 can be 
derived when relation ARC-ATTRIBUTE links with rela- 
tion HIGHWAY-TOPOLOGY by their common attribute 
Are-ID. The details about time operations, historical rela- 
tional algebra for GIS, and extending SQL for historical 
geographic databases are not discussed here. Researchers 
interested in these areas can refer to (Sarda 1990a, Sarda 
1990b, Yang 1991, Yang et al. 1991).