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
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