辅助纬度与大地纬度间的无穷展开

doi:10.11947/j.AGCS.2019.20170747

测绘学报 ›› 2019, Vol. 48 ›› Issue (4): 422-430.doi: 10.11947/j.AGCS.2019.20170747

辅助纬度与大地纬度间的无穷展开

陈成¹, 金立新^2,3, 边少锋¹, 李松林¹

1. 海军工程大学导航工程系, 湖北武汉 430033;
2. 中铁第一勘察设计院集团有限公司, 陕西西安 710043;
3. 甘肃铁道综合工程勘察院有限公司, 甘肃兰州 730000

收稿日期:2018-01-05 修回日期:2018-10-27 出版日期:2019-04-20 发布日期:2019-05-15
作者简介:陈成(1990-),男,博士生,研究方向为地图投影与地球重力场。E-mail:chenchengzyq@163.com
基金资助:
国家自然科学基金（41631072；41574009；41474061）

Infinite expansions of the auxiliary latitudes with respect to the geodetic latitude

CHEN Cheng¹, JIN Lixin^2,3, BIAN Shaofeng¹, LI Songlin¹

1. Department of Navigation, Naval University of Engineering, Wuhan 430033, China;
2. China Railway First Survey and Design Institute Group Co. Ltd., Xi'an 710043, China;
3. The General Engineering Survey Institute of Railways of Gansu Co. Ltd., Lanzhou 730000, China

Received:2018-01-05 Revised:2018-10-27 Online:2019-04-20 Published:2019-05-15
Supported by:
The National Natural Science Foundation of China (Nos. 41631072; 41574009; 41474061)

摘要/Abstract

摘要： 利用无穷级数理论和拉格朗日反演定理，详细推导了大地测量和制图学中常用的辅助纬度与大地纬度间的无穷展开，主要表现为参考椭球第一偏心率的幂级数形式。通过建立一系列严格的系数递推公式，得到了等量纬度反解展开式和等角纬度反解展开式；同时，推导了古德曼函数的泰勒展开式，进而得到了等角纬度正解展开式；利用级数除法公式，得到了等距离纬度正解展开式系数的行列式表示。通过比较本文方法与计算机代数系统Mathematica直接推导求得的辅助纬度正反解展开式e⁰~e⁴⁰阶系数和相应的程序用时，表明本文算法是正确的、快速的。以CGCS2000参考椭球为例，对辅助纬度正反解进行了算例分析，也进一步验证了本文公式的正确性。

关键词: 辅助纬度, 大地纬度, 偏心率, 拉格朗日反演定理, 古德曼函数

Abstract: By using the theory of infinite series and the Lagrange inversion theorem, the infinite expansions of the auxiliary latitudes in geodesy and cartography with respect to the geodetic latitude are given, which are the power series of the first eccentricity of the reference ellipsoid. The inverse solutions of the isometric and the conformal latitudes can be obtained from the recurrences of a number of known coefficients. The forward expansion of the conformal latitude is proposed on the basis of the Taylor expansion of the Gudermannian function, and the forward expansion of the rectifying latitude is given in which the coefficient can be expressed as a explicit determinant by the division of power series. Compared the coefficients of the auxiliary latitudes expansions at order e⁰ to e⁴⁰ with the results by the computer algebra system Mathematica and the corresponding cost time, the algorithm of this paper is verified to be correct and fast. By taking CGCS2000 reference ellipsoid, the numerical experiment is made for the expansions and then the correctness is further proved.

Key words: auxiliary latitude, geodetic latitude, eccentricity, Lagrange inversion theorem, Gudermannian fuction

中图分类号:

P282

陈成, 金立新, 边少锋, 李松林. 辅助纬度与大地纬度间的无穷展开[J]. 测绘学报, 2019, 48(4): 422-430.

CHEN Cheng, JIN Lixin, BIAN Shaofeng, LI Songlin. Infinite expansions of the auxiliary latitudes with respect to the geodetic latitude[J]. Acta Geodaetica et Cartographica Sinica, 2019, 48(4): 422-430.

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辅助纬度与大地纬度间的无穷展开

Infinite expansions of the auxiliary latitudes with respect to the geodetic latitude

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