地貌特征点的数学描述与分析

Mathematical Description and Analysis of Topographic Feature Point

在用等高线表示3维空间地貌形态的科学性这一问题得到解决之后,自然想到如何用数学的方法来描述地貌的各种特征点。地图中的峰、谷、鞍点对应着二元函数的极大值点、极小值点以及马鞍点,首先从地貌函数的二阶方向导数入手给出了判别它们的一个基本条件,并且给出了判别它们的一个较强的充分性条件的简捷证明。然后从概念出发,用函数分析方法首次给出了地性线的宏观定义和微观定义,同时还给出了地性点的科学定义、性质以及与等高线曲率极值点的关系定理和证明。

After the reasonability of the representation of 3D （ three-dimensional space） is elaborated by using 2D contours , how to describe the feature points with the method of mathematics follows nationally. The peak, valley and saddle points in the map are the maximal value, minimal value and saddle points, respectively, in duality function. The paper proved the basic and also the perfect full conditions of determining the feature points starting from the twoorder directional derivative of topographic function. The macro-and micro-definition for topographic feature lines was firstly presented by means of function analysis. The scientific definition of topographic feature points, its characteristics and its relationship with contours curvature extremum points were proposed and proved in the meantime.