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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
among the objects over the map area. Each object has its own 
power as a function of its properties, the surrounding objects, 
map scale, and map type. This power controls the behavior of 
the object in the generalization process in accordance with the 
cartographic rules. 
2.1 Basic Object Parameter 
The basic parameter in the generalization process is the object 
area at the scale of the new map. The object symbol has its 
location, its dimension, and its relative importance among other 
objects, according to the map type. The same objects may be of 
different relative importance when plotted on different maps for 
different uses. 
2.2 Shape Analysis Methods 
Many cartographic objects represented on large scale maps can 
be considered to be geometric objects in the form of polygons 
or closed polylines. These geometric objects could be described 
by the set of shape parameters, consisting of, but not limited to, 
numerical values and topological descriptors (Guienko & 
Doytsher 2003). The shape analysis methods themselves could 
be formally classified bv certain parameters. There are two 
major groups of shape analysis methods: boundary techniques 
(external analysis) and global techniques (internal analysis). 
These methods are applicable to different representations of the 
same object. This paper, however, focuses on the objects as 
polygons in vector format, derived from GIS databases. 
2.2.1 Major Geometric Shape Parameters 
The following major parameters can be used to describe the 
shape of geometric objects in polygon form: area, perimeter, 
centroid, major axes and angle, elongation, compactness, 
solidity and convexity (Guienko & Doytsher 2003). The first 
five parameters could be calculated using moments. Using 
moments for shape description was initiated by (Hu, 1962), who 
proved that moment based shape description is information 
preserving. This study concentrates on the second moment, the 
moment of inertia that can be used to determine the principal 
axes of the shape. 
Moment inertia can be computed in respect to the object shape 
by its vertices or by its edges. Using the length of the edges of a 
polygon rather than its vertices is preferable as it is independent 
of the number and density of vertices and is a function of the 
polygon shape only (Doytsher 1979). Thus, the moment inertia 
computed hereafter is based on edges, with weights proportional 
to their lengths. Therefore, the moment for the two major axes 
in this study is calculated as follows: 
  
S'OG- y.) de, zo NT» 
( Ax si | „y= m dx, mix N 
A) dx, z 
Y (X - x ):dy XoX 
(Qu day ELE ux "Hu . dy, =|y, 83 
AS dy 2 
In order to obtain effective information on the shape of the 
object, the ratio between the moments is also calculated (ratio 
between the larger and the smaller moments). When the 
numerical value of the ratio is close to 1, it indicates that the 
shape is approximately a square, and it is more stable than a 
prolonged shape (with a numerical ratio figure much higher 
than 1). Other parameters like compactness and solidity, can 
also provide useful and important information about the object’s 
shape (Guienko & Doytsher 2003). These parameters are 
calculated as follows: 
; Az - Area ; i i 
(3) compactness = ———————  . perimeter — y sqri(dx” + dy”) 
. t 2 - 
perimeter 
a area 
(4) solidity = 
convexarae 
2.3 Spatial Analysis 
One of the well-known methods of spatial data analysis is 
spatial data mining; a field dealing with producing new spatial 
data from existing data. Data mining is facilitated by utilizing 
shapes or properties which are not explicitly expressed in the 
original databases, and is performed without changing them 
(Kang 1997). Databases kept in GI systems constitute a valid 
potential for data mining due to the vast and diversified data 
stored in them. In this study spatial data analysis is attained by 
implementation of the Delaunay Triangulation and Rings 
analysis. Both methods allow producing large quantities of 
information about the data, density and shape, even though no 
previous explicit knowledge existed regarding these properties. 
2.3.1 Delaunay Triangulation 
Generally, triangulation of a planimetric surface is the method 
of dividing the surface into a finite number of triangles. This 
method is the key tool for handling problems requiring solutions 
based on area division according to the principles of finite 
element theory. 
There are number of methods for performing triangulation, of 
which the Delaunay triangulation is preferred for cartographic 
purposes, since it supplies triangles with the shortest edges. At 
the first stage, a constrained Delaunay triangulation is 
performed in order to include the polygons or line edges, 
describing the objects as part of triangle edges. The triangles in 
the triangulated surface are divided into two groups: 1) triangles 
contained within objects, and 2) triangles stretched between 
objects in the intermediate space (Joubran & Gabay 2000). 
It is the edges of the second group triangles, or their area that 
serve as an indicator of data density in the region surrounding a 
given object. The area of the empty triangles surrounding each 
object and the average length of their edges determine its 
density and the surrounding free area. The surrounding objects 
between which and a given object these triangles are stretched, 
affect the objects density and behavior, and are therefore 
defined in this study as "neighbor objects". 
  
  
  
  
  
Figure |. Sample of Delaunay triangulation - surrounding 
triangles stretched between an object and its neighbors 
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