点线面地图符号的拓扑结构及地图状态变异阐释

Topologic Structure of Point, Line, Area Cartographic Symbols and Explanation of Cartographic State Variation

摘要: 定位点集为零维、1维和2维是点、线、面地图符号的本质差异。从映射(函数)、拓扑变换(同胚)、拓扑空间、连续映射、约当曲线、分离并等的定义出发，阐释地物演化的数学基础。基于图形和背景互为补集的经验事实，推出了地图的表达通式。论证了点线符号集和面状符号集的分离并拓扑结构。基于自然原因和人类活动的地物演化普遍存在，由人类意识行为引发的地物演化是演化的主因。地物演化使制图区域发生初始状态X不同于此后的目标状态Y的状态变异。设X和Y为拓扑空间，$f:X\to Y$ ，因f在X上的每一点连续，所以应用拓扑映射，包括地物扩大或缩小的量变，地物消亡或新生的质变等，可以表达状态变异。用数学变换证明了X与Y状态不同的必然性。实例表明，对地物赋值后，经过有限的布尔代数运算，必能实现从初始状态X到目标状态Y的变换。实例是地图状态变异的数学模型。

Abstract: Location point sets for 0-dimensional points, 1-dimensional lines and 2-dimensional areas constitute the essential difference among point, line and area map symbols. Starting from the definitions of mapping (function), topological transformation (homeomorphism), topologic space, continuous mapping, Jordan curve, disjoint union of sets, etc., this paper explains the mathematical foundation of geo-object evolution. The general formula for map representation is derived based on the empirical fact that figure and background are mutually complementary. The topological structure of the disjoint union of point-line symbol sets and area symbol sets is demonstrated. Geo-object evolution driven by natural factors and human conscious activities exists universally Geo-object evolution triggered by human conscious behavior is the primary cause. Such evolution can produce state variation: the initial state X differs from the subsequent target state Y. Let X and Y be topological spaces, $f:X\to Y$ , Since f is continuous at every point on X, topological mapping can be used to express state variation, including qualitative changes such as expansion and contraction, and qualitative changes such as disappearance and emergence Mathematical transformation proves the inevitability that the initial state X differs from the target state Y. Examples show that after assigning values to geo-objects. The transformation from initial state X to target state Y can certainly be realized through finite Boolean algebraic operations. These examples are mathematical models of map state variation.