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

doi:10.11947/j.AGCS.2019.20170747

Acta Geodaetica et Cartographica Sinica ›› 2019, Vol. 48 ›› Issue (4): 422-430.doi: 10.11947/j.AGCS.2019.20170747

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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)

Abstract

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

CLC Number:

P282

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

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