THE BASIC TOPOLOGY MODEL OF SPHERICAL SURFACE DIGITAL SPACE 
HOU Miao-le! 
' Department of Surveying & Land Science, China University of Mining and Technology (Beijing), D11 
Xueyuan Road, Beijing, China, 100083 
e-mail: hmlicumtb.edu.cn 
Keywords: SGDM, manifold, spherical surface digital space. spherical surface digital topology, spherical surface digital image, 
spherical surface spatial relationship 
Abstract: 
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. What's more, 
most of mathematical model about global application have been based on the continuous methods. Although several researchers have 
considered the digital topology in 2-dimension and 3-dimension Euclidean Plane, complete theoretical foundation of proper theory for 
spherical surface digital space is still missing. In fact. it’s more convenient and efficient to compute spatial relationship based on 
spherical surface digital space. 
Firstly, this paper constructs spherical surface digital space based on manifold, ie. the digitization of the spherical surface as a 
common spatial framework just as planar. Secondly, the concept of spherical surface digital topology will be given in the context of 
; ; 2 . < S 2 
adjacency. We define a spherical surface digital image FP asa triplet (7 À, H) , where Æ isa finite subset of 7° and R 
represents the adjacency relation in the whole lattice in a specific way. Then topological properties and paradox of discrete objects will 
be discussed in spherical surface digital space. In the end, we give some potential applications about spherical surface digital topology. 
just because of the advantages of multi-resolution and 
: hierarchy. 
I. INTRODUCTION = 
Computers are digital, and most image acquisition and 
communication efforts at present are toward digital approaches. 
However, vector model including direction cosine and longitude 
and latitude coordination are continuous in essential. There 
must be some deficiency between continuous model and 
The surface of the earth is an important spatial domain, which is 
of course topologically equivalent to the surface of a spherical 
surface, ellipse and geoid. In fact, it's not topologically 
equivalent to any subset of the Cartesian plane (Sahr 1996 
White 1992, 1998). So it's unpractical to analyze the spherical discrete computation just as figure 1. SGDM is a promising 
method to deal with the spherical surface. Therefore, it's 
surface with the planar methods. Three typical conceptual i 
conveniently to analyse the global data and make decision in 
modeling approaches are used to apply and adapt spatial 
analysis techniques to the spherical surface as follows (Raskin spherical surface digital space. 
1994). 
® Map projection: The spherical surface is projected onto a 
plane using a conventional map projection. The projection Discrete Computation 
approach is used implicitly in conventional studies that AV 
ignore the curvature of the earth, however, no one „x. 
projection can keep both distance and area. What’s more. 
the map projection transforms the spherical surface 
manifold to planar Euclidean space, therefore, the X 
distance, orientation and area in large field are not ot’ 
accurate at all. Real World + 
® Embeddings: The spherical surface is considered a 
constrained subset of the three-dimensional space R°. 
Tree-dimensional spatial analysis is performed, with a (a) 
constraint imposed to limit solutions to the spherical 
surface. The most typical one is direction cosine, which 
avoids the singularity of the pole. Although direction 1 : mus 
= € Discrete Computation 
cosine has a perfect mathematics base, so it belongs to the T 4 
.** 
* * 
  
  
  
  
  
  
Model Euclidean space Discrete Data 
  
  
  
  
  
  
  
  
The relation between vector model and real world 
  
  
  
  
vector method and does not accord with the discrete 
  
  
properties of real word in essential. T 
9 Intrinsic: The spherical surface is considered an intrinsic Model Digital space imc Discrete Data 
  
  
  
  
  
space in its own right, with analysis performed in 
non-Euclidean space. S^. Longitude and latitude 
coordinate and SGDM are most typical two. SGDM 
(Sphere Grid Data Model) is research topic of this paper 
  
Real World