依比例符号、不依比例符号和半依比例符号数学定义的改进

Improving the Mathematical Definition of Scale Symbol,Non-scale Symbol and Semi-scale Symbol in the Map

三维空间中的制图物体要投影到地球椭球面S上,故存在三维空间X到地球椭球面S的映射f:X→S;x及其投影f(x)的属性、尺寸、形状等特征,被制图者认识和了解是对x进行地图表示的前提,故存在椭球面S到主体认识结构Y的映射g:S→Y。x的观念模型gf(x)可以表示在二维平面Z上,故Y与Z间必然存在映射q:Y→Z。根据非退化区间为等势集合的定理,知f、g、q为双一一函数,从而使x、f(x)、gf(x)与qgf(x)一一对应。而f(x)与qgf(x)是否与地图比例尺1∶M相关、无关或半相关,决定着地图符号qgf(x)为依比例符号、不依比例符号或半依比例符号的归属。这三类地图符号相互转化及条件,证明了事物由量变到质变的辩证规律。

The mapping objects in the three-dimensional space need being projected on S surface of earth-ellipsoid, so there is the mapping f： X→S from three-dimensional space X to the S surface of earth-ellipsoid. It is the premise of showing x on map for the attribute, measurement, shape, etc. characters of x and its mapping f（x） on S are known and undersdood by the cartographer, so there is the mapping g：S→Y from the S surface of earth-ellipsiod to the body cognition construction Y. The idea model of x in Ygf（x） may show on twodimensional plane Z, so there is the mapping q：Y→Z from the body idea construction Y to the two-dimensional plane Z. According to the theorem non-deteriorate inter-regional belonging to the equipollence set, we know that the f, g, q are bijective, this makes x, f（x）, gf（x） and qgf（x） corresponding one to one. It decides the map symbol qgf（x） belonging to the scale symbol or non-scale symbol or semi-scale symbol respectively for the f（x） and qgf（x） with the map scale 1 ： M are related, aren＇t related or semi-related.