4. Istanbul 2004 
pherical surface 
xterior. A point 
Nip) CF, 
The set of all 
of E and is 
E is called the 
osure of E is 
jor, closure and 
definition of 
> complement of 
| plan 
nents. This 
tor space and it 
r space. So a 
). Kong and 
Ived if the white 
k spels 
(from LI et al. 
a 
paradox in T5. 
y spel and some 
six black spels. 
? connected and 
ack line cannot 
white spels. If 
arate the central 
' black spels are 
| been formed by 
the topological 
his paradox, the 
cted and black 
pherical surface 
e the different 
tity in spherical 
1e background 1s 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
defined as being 3- connected . So, the six black spels 
defined as 12 - connected should be connected. However, 
gray spel and black spels just as background should be not 
connected if the background is defined as being 
3- connected . So the continuous curve (connected path in 
2. . 
T separate the spherical surface two parts. 
         
N 
d NU N Z Nod f \ 7 = 
Figure 9. Topological paradox in raster space 1” 
  
But why this topological paradox happens? We use the six spels 
in figure 10 to explain it. In reality, when one considers spels 
1.3 and 5 to be connected, one has already implicitly assumed 
that P belongs to the black line. On the other hand, when one 
considers spels 2, 4 and 6 to be connected, one has already 
implicitly assumed that P belongs to the white spels. That is, the 
point P belongs to two different things. If the black spels 
represent spatial entities and the white spels represent the 
background, then point P belongs to both the background and 
the entity at the same time, thus having dual meanings. This of 
course leads to paradox-a kind of ambiguity. To solve the 
problem, one must eliminate the dual meanings of point P. One 
should only allow P to belong to either the entity or the 
background but not both. In this paper, the spels belonging to 
background are defined as 3-connected, however, the spels 
belonging to the object are defined as 12-connected. 
VEDI LIS 1 E 
SHE 
  
Figure 10. Topological paradox caused by the ambiguity at 
point P. 
3.4 Relationship between topologies on spherical surface 
digital space and spherical surface continuous space 
The connectedness of raster space is based on the adjacency of 
two neighboring spels (LI et al. 2000). In spherical surface 
digital space, there is a common line (Figure 11a) between the 
two spels in the case of 3-connectedness. On the other hand, in 
the case of 12-connectedness, the common part could be either a 
line, a point, or both. In other words, there is at least a point in 
common if the two spels are to be connected. If an arbitrary 
(vector) point is selected from cach spel, say "a" and "b", then 
the path from "a" to "b" intersects the common line at P. Points 
"a", "b" and P are points in vector space. Points "a" and P are 
connected in the left spel and points P and "b" are also 
connected in the right spel in vector space. As the connectedness 
is transitive, points "a" and "b" are therefore connected. As a 
result, any point in the left spel is connected to any point in the 
right spel. It means that the connectedness concept in vector 
space has been implicitly adopted when the connectedness 
concept in raster space is discussed. 
  
(a) in the case of 3-adjacency 
  
(b) in the case of 12-adjacency 
Figure 11. Implicit dependency of topological connectedness in 
e 
T 
CONCLUSION 
SGDM (Sphere Grid Data Model) is an efficient method 
to deal with the global data because of the advantages of 
multi-resolution and hierarchy. However, SGDM has no distinct 
descriptions and lack of round mathematical basis for various 
applications. This paper gave the definition of spherical surface 
digital space, which has the characters as follows: 
9 Similar regular grids based on spherical surface 
discrete space. 
€ Spherical spacefilling curves can be used to express 
the relationship between local coordination. 
€ No single coordination system can express every 
point in the spherical surface. 
€  Multi-scale and continuous ordering. 
As an important part, this paper set up the basic topology model 
which include the topological structure of sphere digital space, 
2 
the basic topological components of a spatial entity in. T^, 
topological paradox associated with definition of adjacency in 
T° and so on. This paper 1s just an introduction to studying 
the characterization of 2-digital sphere manifold and the 
Jordan-Brower separation theorem, which are all round 
mathematic basis of spherical spatial computing and reasoning.