• 大地测量学与导航 •

### 辅助纬度与大地纬度间的无穷展开

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 Lixin2,3, BIAN Shaofeng1, LI Songlin1

1. 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)

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.