object’s spatial states and thematic states. A state line in 
Figure 2.1 may be viewed as the composition of the 
object’s parallel spatial state lines and thematic state lines. 
For example, suppose the object has a point feature, such 
as gas station A, shown in Figure 2.2. At time TO, station 
A was created. At time T1, the station was moved to a 
new location. At time T2, the ownership of the station was 
changed, and the new owner moved the station to a new 
location at time T3. Finally, at time T4, the station was 
closed. Figure 2.3 shows the state topology of station A. 
Another example is shown in Figure 2.4. Polygon A 
represents grass-land and was created at time TO. At time 
T1, a part of grass-land A became farming land (arc al 
was changed into arc all). At time T2, all of farming land 
B became grass-land (arc a2 was removed, and arc a5 
became an arc of polygon A). At time T3, polygon D was 
split into two polygons, and caused arc a4 to be split into 
arc a41 and arc a42. At time T4, the ownership of grass- 
land A was changed. And finally, at time T5, all of grass- 
land A became a commercial area, and was combined with 
commercial area C. This means that grass-land A does not 
exist any more. The state topology of polygon A is shown 
in Figure 2.5, where we assume that each arc has only one 
state. That is, any change of the arc is viewed as the death 
of the arc, and the birth of zero or more new arcs. 
  
  
  
  
  
  
  
  
  
  
  
  
  
TO TI 
T2 
  
T3 T4 
Figure 2.2 The Mutations of Gas Station A 
State Mutation 
   
  
The Spatial States 
of Station A 
  
  
M, y T 7 7 7 ^ ^ ^ ^ ^ ^ 
  
'The Thematic States 
of Station A 
The States of 
Station A 
  
A eu vi ut cie 
I 
i 
TT 
i 
1 
1 
  
dj 
Figure 2.3 The State Topology of Gas Station A 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
a8 a5 
B 
E 
a2 
all 
A 
a4 a3 
D C 
a7 a6 
T0: Polygon A Created T1: Arc al changed 
into arc all. 
a8 a5 a8 a5 a8 a5 
E E E 
all A all A all C 
j a41 a3 a41 
a4 a3 a7 D a7 D 
D C a42 C a42 
D1 DI 
a7 a6 a71 a6 a71 a6 
  
T3: Arc ad split into arc a41 
and a42 
T2: Arc a2 was removed and Arc T5: Polygon A died 
a5 became an arc of polygon A 
Figure 2.4 The Spatial Changes of Polygon A Over Time 
157