地图投影计算机代数分析研究进展

Research Progress in Mathematical Analysis of Map Projection by Computer Algebra

地图投影是现代地图学的重要组成部分,涉及大量的椭圆函数幂级数展开、隐函数复合函数微分、椭圆积分、复变函数运算等一系列烦琐的数学分析过程,人工推导不但费时费力,而且容易出错,有时由于难以忍受的复杂性等各种原因,甚至根本无法实现。本文主要从椭球各纬度间正反解符号表达式、不同变形性质地图投影间的直接变换、高斯投影的复变函数表示、斜轴墨卡托投影数学分析、极区海图投影及变换等5个方面,论述了地图投影计算机代数分析取得的研究进展,讨论了该领域有待进一步解决的主要问题,对推动地图投影学的发展具有积极意义。

Map projection is an important component of modern cartography,and involves many fussy mathematical analysis processes,such as the power series expansions of elliptical functions,differential of complex and implicit functions,elliptical integral and the operation of complex numbers.The derivation of these problems by hand not only consumes much time and energy but also makes mistake easily,and sometimes can not be realized at all because of the impossible complexity.The research achievements in mathematical analysis of map projection by computer algebra are systematically reviewed in five aspects,i.e.,the symbolic expressions of forward and inverse solution of ellipsoidal latitudes,the direct transformations between map projections with different distortion properties,expressions of Gauss projection by complex function,mathematical analysis of oblique Mercator projection,polar chart projection with its transformation.Main problems that need to be further solved in this research field are analyzed.It will be helpful to promote the development of map projection.