International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
e Ifa b. € k and ab = D, then ab is a tcell and a 
union of elements of k where tcell is either a or b (of 
different dimension) or a common face. 
The discussion is restricted to n = 2 (2D space-time), where m 
€ {0,1.2}. Therefore, R"” represents the object in E" at time t, 
where m is the dimension of the object and z is the dimension 
of space where the object is located at time t, such that 5 2 m. 
As defined earlier, ZeroTCellClass, OneTCellClass, and 
TwoTCellClass are the subclasses of SpatioTemporalClass. In 
this paper we shall focus on TwoTCellClass. The object of this 
class is a TTC. A TTC is a two-dimensional object bounded by 
a closed cycle of ZTCs and OTCs. Any j-th TTC = set{OTC,, 
OTC, OTC, OC, where 1 = 11.2, n}. When 1 = |, 
the first and last point object and the start and end ZTC are 
identical. 
The data members of TwoTCellClass are a set of onetcells, 1-T 
(systemtime), area, perimeter, and parent (TwoTCell). The life 
of each TTC is depicted by 1-T [Troms Tuna]. Like OTC, this 
object too can either be born or die. The birth and death times 
are represented by two point times,  Trom. 20d. Tun 
respectively. A TTC must have one or more OTCs, and an 
OTC may have zero or two TTCs. Each TTC object may have 
one or more children, and each child (TTC) must have a parent 
(TTC. 
When an n-tcell can be born or can die is an important decision. 
It is logical to investigate the situations where an z-tcell is 
changed as a result of the updating (insert or destroy) process in 
the spatiotemporal database. These operations are necessary for 
designing the algorithms of various operators. 
While updating the data, an object (7-tcell) can intersect with 
another object. In the unified spatiotemporal data model, when 
a ZTC, OTC, or TTC is inserted, the following scenario can be 
expected: 
e A ZTC may intersect with ZTC, OTC, or TTC. 
e An OTC may intersect with ZTC, OTC, or TTC. 
e A TTC may intersect with ZTC, OTC, or TTC. 
There are nine possibilities when an »r-tccll at time TI may 
intersect with an n-tcell at time T2. In each case, there are 
various possibilities (e.g., ZTC may intersect at the boundary of 
OTC or the interior of OTC). Cases in which a TTC intersects 
with a TTC are discussed in the paper. The three topological 
invariants of spatial objects (n-tcells) are boundary, interior, 
and exterior. 
This point-set topology approach is employed to analyze these 
intersections. Only the boundary (0) and interior (?) of OTC 
and TTC are considered in order to investigate these 
intersections. The intersection at the exterior of any 7-tcell is 
straightforward. The boundary of ZTC is empty (©). A similar 
approach (Le., point-set topology) has been employed by 
Egenhofer ef al. (1994) to identify and/or compare topological 
relationships between n-dimensional objects embedded in R^. 
3. TEMPORALCELLTUPLECLASS 
Because the object is defined as a spatiotemporal object, the 
topological relations could be defined as spatiotemporal 
topological relations (i.e., the spatial relations that are valid 
over time). In the temporal cell complex, /ntra cell complex 
relations (i.e., relations between the cells in the cell complex) 
can be described using boundary and co-boundary relations. 
The boundary (8) of an n-tcell is its (5-7) faces at time t. The 
co-boundary (®) of an n-tcell produces the (n^ 7) cells incident 
with »-tcell at time t The boundary and co-boundary relations 
capture two types of topological relationships (i.e., adjacency 
and containment). Relations between spatial objects can be 
found based on boundary/co-boundary relations between cells. 
The boundary and co-boundary relations are encapsulated in a 
simple temporal cell tuple structure, which is an extension of 
the cell tuple structure of Brisson (1990). A cell tuple T is an 
(n+1)-tuple of cells {co, Ci, C2,-Cnts where any i-cell is 
incident with a (7+/)-cell. 
The object of TemporalCellTupleClass has a unique tuple-ID 
and a unique combination of ZTC, OTC, and TTC. Each tuple 
must have a ZTC, zero or one OTC, and zero or one ic. 
Therefore, a temporal cell tuple structure encapsulates the 
spatiotemporal topology of cach spatiotemporal object. A 
temporal cell tuple (TCT) is a set of C and T. 
Ter =16 31 
where C is a set of cells 
C= fe, oil erin. Chl oe TCC and 
T is a time interval (1-1) 
I= VT rom Tt nul | (T rom << Tunii) ^ CT ion d'en € ST) 
and 
TCC = TemporalCellComplex 
Therefore, 
TE = (Co €, C5,.... Co Tro loni] 
The process of assigning the cell tuples to a 71€ 
illustrated in Figure |. 
n 
a 
c (n,0,0,1-T) 
0 
(al 0 
ct (n1, 0. A, 1-T) 
c2 (n2, a1, A, 1-T) 
c2 (n2,a1,0,1-T) c3 (n2, a1. 0, 1-T) 
ad n2 c4 (n3, a1, A, 1-T) 
nl det c5 (n3, a1, 0, 1-T) 
P c6 (n3, a2, A, 1-T) 
c1 (n1,a1,0,1- T) c7 (n3, a2, 0, 1-T) 
0 c8 (n2, a2, A, 1-T) 
c9 (n2, a2. 0. 1-T) 
[b] [c] 
Figure |. Process of assigning temporal cell tuples to 
spatiotemporal cells of dimensions (0 € n € 2). 
4. OPERATORS 
Two types of operators can be defined (ie, static and 
dynamic). Static operators do not affect the system's state or the 
status of spatiotemporal objects (e.g.. query operators calculate 
the length. area, time period, boundary, or co-boundary). These 
operators are associated with TemporalCellTupleClass. On the 
other hand, dynamic operators change the state of the system or 
the status of the spatiotemporal objects (e.g.. creating, deleting, 
or updating an z-tcell). Normally in atemporal GIS, three 
fundamental dynamic operations are performed {i.c., Create, 
delete, and update) These operators are associated with 
PointClass, ZeroTCellClass, OneTCellClass, and 
Intei 
  
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