Mathematical Definitions of Point，Line and
Area Symbols in Cartogtaphy

WEI Wenzhan，Zhong Yexun，PENG Yueying，ZHENG Hongbo

【ABSTRACT】 The line cartographic symbol located on the boundary and the point cartographic symbol located by single point are the main body and stress of the map. The cartographic symbols are divided into point，line and area symbol in cartography and traditional educational books，but upto now there is only qualitative description about the three kinds of symbols，the are not the mathematical definitions of point，line and area symbols. This paper tries to solve the problem. The authors propose the mathematical definition about point，line and area cartographic symbol according to the theory of topology and expound their dependence relation.
【KEY WORDS】 point cartograrphic symbol；line cartographic symbol；area cartographic symbol；figure

　　1 Introduction
　　The are visible or invisible boundaries between things and phenomena of various features in cartographic region. The boundaries not only indicate the borders of objects，but also effectively reflect the objects on the borders. Therefore，the line cartographic symbols located at the boundaries and the point cartographic symbols located at single points are the main body and stress of the map. The cartographic symbols are divided into point，line and area symbols in cartography and traditional educational books. Up to now，there is only qualitative description about the three kinds of symbols but the mathematical definitions of point，line and area symbols. This paper tries to make up for this.
　　2 Point cartographical symbol
　　The important characters of point cartographic symbol are that it locates as a definite coordinate point C（x，y） and indicates the feature or some quantitative values on the point. The point cartographic symbols are related to more visual variables such as shape，size，direction，structure，color etc.^[1] So the point cartographic symbol belongs to the point set category.
　　Definition 1　 There exists planar coordinate point C（x，y），if the following condition is satisfied：

（1）

　　then Ui is called the point cartographic symbol，which is fixed on C（x，y） and indicates the feature i.
　　There are many kinds of fixed means of point cartographic symbol（Fig.1）. In various fixed means the positions of fixed point C（x，y） in Ui are different. But all of them satisfy the condition that C （x，y） ∈Ui ，Ui=f（i） and Ui indicates the feature of I on the map.

Fig.1 Sketch of point cartographic symbol

　　Eq.（1） is the common equation of point cartographic symbols such as triangle point，tower building，transformer substation，single tree etc.
　　3 Line cartographic symbol
　　There exist the borderlines among different characteristic and different extent features. The line cartographic symbol is fixed on the borderling and indicates the function of the feature，so we must definite the neighborhood and boundary （Fig.2）

Fig.2 Relation of neighborhood，boundary and line cartographic symbol

　　2.1 Neighborhood
　　Definition 2 　Let a be a point in metrical space X and Na be a point set in X，if the following condition is satisfied.

（2）

　　then Na is called a-neighborhood in X.
　　2.2 Boundary
　　Definition 3 　Let A be a subset in topological space X，a∈X，if an arbitrary a-neighborhood has points in both A and X-A，then we call a is a bounding point of A. The set of all bounding points is called the boundary of A，which is denoted by BdA^[3]，and it can be written as

　　2.3 Line cartographic symbol
　　Definition 3 　Let A be a subset in topological space X，α∈X，if an arbitraryα-neighborhood has points in both A and X-A，then we call α is a bounding point of A. The set of all bounding points is called the boundary of A，which is denoted by BdA，and it can be written as

　　2.3 Line cartographic symbol
　　Definition 4 　The cartographic symbol fixed on a related boundary is called line cartographic symbol which is a point set，and indicates the line feature i . It satifies the following condition：

（3）

　　Here，BdA is provided by Eq.（2） and i is the feature indicated by line cartographic symbol. Eq.（3） is the common equation of line cartographic symbol such as single line river，road，border etc.
　　4 Area cartographic symbol
　　3.1 Connected set
　　Definition 5 　There exists a point set A∈X，if A can not be expressed by separated sum aggregate of two non-null open sets，namely，there can not exist open subsets U，V of A，U≠Φ，V≠Φ，i，e，A=UЦV，then set A is called connected set^[2].
　　3.2 Area cartographic symbol
　　Definition 6 　If there exists connected set Aj∈X，which indicates the attribute of feature surrounded by the boundary BdAj of Aj. Then point set Aj is called the area cartographic symbol and satisfies

（4）

　　Here，Aj is provided by Definition 5，and j is the frature indicated by area cartographic symbol. Cinsidering the different essence of point，line and area cartographic symbol，we agree on that I indicates point and line feature，and that j indicates area feature，Eq.（4） is the common equation of area cartographic symbol such as water field，woodland，plowland etc（Fig.3）. Because the connected set has complete boundary，it can describe any area field.

Fig.3 Sketch of area cartographic symbol

　　5 Figure and background of map
　　4.1 Point and line cartographic symbols are the main body of map
　　1） Map is the subset of graph is the basic condition and logical requirement of cartographic definition^[4]，and the set of point and line is the main condition of constructing graph^[5].
　　2） The map load is composed of elemental symbols and lettering areas which settle place，drainage，road network and boundary lines，etc.^[6] ，i.e.，it is composed of point and line symbols and its lettering area. The map load is concerned with the map legibility and map quality^[7].
　　3） Point and line cartographic symbols always lie on higher visual level in the map. The important point and line symbols are more significant.
　　4） Constructing the graph of point and line symbols is the comparative standard when interpreting the map belonging to negative images or positive images，right-reading or worng-reading ^[8]. Constructing the graph of map’s main elements expresses the sum aggregate which are the point symbol set{Ui} and the line symbol set 　　{li}，i ，e.

（5）

　　4.2 Area cartographic symbol constructs the background of map
　　Definition 7 　The map’s ground is the sum aggregate of the whole area cartographic symbols，or it is complement of sum point and line symbol sets on the map. i .e.

（6）

　　where Aj is provided by Eq.（4） and G is provided by Eq.（5）. Any map is always the unitive body of figure and background. Eq.（6） expresses this relation of opposite and union.
　　4.3 The figure and ground play the whole function in the transmission of map imformation
　　The map is the symbolic model of the objective world. The research shows that the information in map is more abundant than than that ingeneral books and newspapers ^[9]. Nowadays，a general map can contain and store 100-200 million pieces of information ^[9]. Therefore，as the media and bridge of human being recognizing the objective world，map is more intuitional and more effective than other means. Though the figure and the background lie on different visual levels，they perform united function in transmitting geographic information for their interdependent relation. They perform a mutually complementary function for reflecting the geographic characters.①Unformed or inconspicuous form borderlines in the objective world are prominent for the structure of point and line symbols，and the area symbol is prominent for the prominent borderline and the difference of borderline and area symbol in vision.②The area cartographic symbols as light color can make difference each other between the point and line map symbols of different quality，thus various point and line symbols are effectively reflected in the map.　Especially，recessive things （such as air temperature，border，population） are displayed and emphasized by cartographic symbols and we can observe all of them in the map.
　　6 Conclusions
　　On the basis of the theory of set and topology，combining the common attribute and character of cartographic symbols in a mathematic way，which makes confined qualitative description cartographic symbols obtain an exact mathematic form and quantitative description cartographic symbols obtain an exact mathematic form and quantitative description. The interdependent and opposite and union relations of point，line and area cartographic symbols make them produce the important effect and perform united function in the transmission of geographic information.

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