Mathematical Definitions of Operators
for Cartographica Generalization

ZHONG Yexun　　WEI Wenzhan 　HU Yuju　　ZHENG Hongbo

【ABSTRACT】 This paper puts forword 11 cartographic generalization opreratormodels and introduces their mathdefinitions，and thus a precise mathematical form and quantitative description has been given to these formerly limited qualitative concepts. The meaning of mathematical definition of operators for cartograghic generalization and the application prospect in computer-aided cartography （CAC） is stated.
【KEY WORDS】 operators for carto-graphic generalization；mathematical definition；selection；simplification；stress

　　1 Introduction
　　A map shows the related things of the cartographic region in a limited area by using symbol sets. This is the transformation course from geographical information in three-dimensional space into the map information on two-dimensional plane. According to the cartographic aim， different characters or capacity can be given on each point， such as height， temperature， population， etc. Because of the existence of one to many function relations， the map is the homomorphism model of the cartographic region. In order to construct a cartographic model for a certain aim， the category features and typical characters of cartographic phenomena need to be shown by means of condensation and abstraction. And at the same time those secondary and non-essential objects should be abandoned. This process is called cartographic generalization. Many scholars have put forward various resolving models， which include the most common models of three operators （selection， abstraction and movement）and models of four operators（selection， abstraction， mergence and movement）. G.Hake raised models of seven operators in two kinds （pure geometric generalization：simplification， exaggeration and movement； and geometric/conceptual generalization：selection， categorization， mergence and stress）. In GIS community， there are twenty operator models. However， in cartographic literature there are not any strict mathematical definitions for the above stated generalization operators yet. Considering the mutual relation and the logical steps among cartographic operators and the facts of general existence of cartographic generalization and the application， the authors put forward models of 11 operators， namely：selection， abandon， derivation， augmentation， simplification， exaggeration， mergence， division， movement， categorization and stress.
　　2 Mathematical definitions for cartographic generalization
　　2.1 Selection
　　For point set A and selection threshold ， if the following condition

Q1={x∣x∈A∧x≥} 　　（1）

　　is satisfied， then Q1 is a set of selected elements.
　　The computation of selection can be expressed as （see Fig.1）

A'=A∩（x1∪x2∪…）∣x∈A∧x∈A'∈ Q1 　　（2）

　　2.2 Abandon
　　For point set A and selection threshold ， if the following condition

Q2={x=∣x∈A∧x≥} 　　　（3）

　　is satisfied， then Q2 is a set of abandoned elements.
　　The computation of abandon can be expressed as[4] （see Fig.2）

Fig.1 　Diagrammatic sketch of selection transformation


Fig.2 　Diagrammatic sketch of abandon transformation

　　The result of selection can be obtained from selection and abandon operations. A mutual supplementary relation exists between selection and abandon.
　　2.3 Derivation
　　For point set A and abandoned elements set Q2 ， if x∈A∧x∈Q2g（x） A， then g（x） is the derivation caused by the abandon of x.
　　The phenomena of derivation exists universally in maps. In map compilation， the abandon for the menandering and the substitution by straight segments are a pair of operations that can often be seen for rivers， outlines of objects and seashores， where the abandoned zigzag part is x， and the substituted part is g（x）. The appearance of any new objects is accompanied by the disappearance of some old ones. The disappeared objects are abandoned in the classification of objects， while the newly appeared ones are the derived objects.
　　2.4 Augmentation
　　Let xA， then the transformation of xA is called augmentation. The computation of augmentation is expressed as

A'=A∪x∣xA∧x∈A'　　 （5）

　　Where x is the augmentation element.
　　In renewal of maps， the augmentation of new objects belongs to augmentation transformation.
　　2.5 Simplification
　　 Q1，Q2∈A，x∈Q1x≥，x∈Q2 x≥，where is the threshold of selection. Let g（Q2） be the derivation of set Q2 of abandoned elements， if the following condition

A'={x∣x∈Q1∧x∈g（Q2）}，Q1，Q2∈A∧Q2A'　　　 （6）

　　can be satisfied，then A' is a simplification or generalization of A.
　　The computation of simplification is expressed as （see Fig.3）

A'=A∩Q1∪g（Q2）∣Q1，Q2∈A∧g（Q2）A 　　　　（7）

　　The simplification of rivers， coastal lines and contour lines belongs to the simplification transformation. The essence of simplification is a complex computation of selection， abandon， derivation and augmentation.

	the selected elements 　the abandoned elements   　the derivated elements
Fig.3 　Diagrammatic sketch of simplification transformation

　　2.6 Exaggeration
　　For point sets A and B， A∩B=LAB∧B∈Ext A， if the following condition

A'=A∪LAB∪B∣LAB =A∩B∧B∈Ext A　　　　　 （8）

　　can be satisfied， then A' is the exaggeration of A. （see Fig.4）

	LAB 　exaggerated 　boundary
Fig.4　 Diagrammatic sketch of exaggeration transformation

　　2.7 Mergence
　　For point sets A and B， A∩B=， if the following condition

Q={P∣P∈A∧P∈B，P1∈A，P2∈BNP1∩NP2　　　　（9）

　　can be satisfied， then Q is the mergence of A and B.
　　Mergence is a transformation of two separated point sets to a crossed set Q. （see Fig.5）

	______ and _ _ _ _ form 　the boundary of Q， the 　boundary of A and B 　vanished
Fig.5 Diagrammatic sketch of mergence transformation

　　2.8 Division
　　For point set Q， P1，P2， ∈QNp1 ∩ Np2， if point sets A， B and C can be obtained after transformation and fulfill the following condition

A∩C=∧IntB=ExtA=ExtC∣A∪B∪C=Q 　　（10）

　　Then B is the division of point set Q. Two separated point sets A and C are formed after division. Division is the inverse transformation of mergence. （see Fig.6）

	..........the abandoned boundary 　___________ the selected boundary 　 　_ _ _ _ _ _ the augmented boundary
Fig.6 Diagrammatic sketch of division transformation

　　2.9 Movement
　　For point sets A and B， B1 B2 B ，B1∩A=LAB，after A is exaggerated to A'， let B2∈Int A'，and f（B1） is the image of B1 and can fulfill the following conditon.

F（B1） ∈ExtA'∧f（B1）∩A'=LA'B1'　　　 （11）

　　then f（B1） is the movement of B1； B2 is the measuement of movement.（see Fig.7）

LAB1 　　LA'B1'
Fig.7 Diagrammatic sketch of movement transformation

　　2.10 Categorization
　　Let {Xi}i∈I be a subset family of a known set X， I， e， i∈I ，XiX，if the following conditions

　　can be satisfied， then the set family {Xi}I is called a category of set X， and each Xi is a member of the category {Xi}i ∈ I[5].
　　If x， x'∈X， let x and x' belong to the same category Xi. This process is called categorization.
　　In the past， on the middle-scale maps， China’s highways are classified as express highway， general highway， simply constructed highway and highway under construction. While on the small-scale maps， they are divided into main and secondary highways. Obviously， when highways are transformed from middle-scale into small-scale maps， they should be processed by categorization[6].
　　2.11 Defintion 11：Stress
　　Suppose that the selected subset Q1 and abandoned subset Q2 belong to the point set A， and F is the characteristic point set of landforms of some objects if x∈F∧x∈Q2， then x is taken as a selected element rather than an abandoned one. In this case， the process is called a stress of x.
　　Stress is a selection which can satisfy the special condition x∈F∧x∈Q2. At the same time， it is accompanied by some exaggeration and movement. Therefore， stress is a complex computation of selection， exaggeration and movement.
　　3 Conclusions
　　A new map is a working result for a certain goal according to the generalization principle and rules and is performed on the base material. In this paper eleven cartographic generalization operators have been suggested and their mathematical definitions are deduced， and thus the concepts which were formerly limited in qualitative description have got their precise mathematical form and quantitative description. Furthermore， the theoretical summarization and explanation are given. The mathematical definitions for cartographic generalization operators will benefit the CAC software research and exploitation.

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