Symbolic Expressions of Differences between Earth Radii

doi:10.11947/j.JGGS.2020.0105

测绘学报（英文版） ›› 2020, Vol. 3 ›› Issue (1): 45-51.doi: 10.11947/j.JGGS.2020.0105

收稿日期:2018-07-10 接受日期:2018-09-21 出版日期:2020-03-20 发布日期:2020-03-09

Symbolic Expressions of Differences between Earth Radii

Jingwen ZONG¹,Houpu LI¹(),Shaofeng BIAN¹,Qinghui TANG²

1. Department of Navigation, Naval University of Engineering, Wuhan 430033, China
2. PLA of No.32022, Wuhan 430033, China

Received:2018-07-10 Accepted:2018-09-21 Online:2020-03-20 Published:2020-03-09
Contact: Houpu LI E-mail:lihoupu1985@126.com
About author:Jingwen ZONG(1995—),male,PHD, majors in geodetic surveying. E-mail: zjw19950613@163.com
Supported by:
National Natural Science Foundation of China(41771487);National Natural Science Foundation of China(41971416);National Natural Science Foundation of China(41871376);Natural Science Foundation for Distinguished Young Scholars of Hubei Province of China(2019CFA086)

摘要/Abstract

Abstract:

A systematic and comprehensive comparison of the five commonly used earth radii in geodesy and cartography is carried out, and the differences between the most common points of the earth radii, their corresponding maximum values, and the latitudes of equal points between them are derived with the help of computer algebraic systems. The symbolic expressions are expressed as a power series of the first eccentricity. Taking the CGCS2000 ellipsoid as an example, the differences between the commonly used earth radii are clarified to numerical values. The results show that the difference between the commonly used earth radii has a maximum at 90 degrees and a minimum at 0 degree. The difference between the average radius of curvature and the rectifying sphere radius is the biggest, and the difference between the average radius of curvature and the average sphere radius is the smallest. These results can provide a theoretical basis for corresponding research in the geosciences, space science, navigation and positioning.

Key words: earth radius, difference extrema value, extrema value point, symbolic expression, CGCS200

. [J]. 测绘学报（英文版）, 2020, 3(1): 45-51.

Jingwen ZONG,Houpu LI,Shaofeng BIAN,Qinghui TANG. Symbolic Expressions of Differences between Earth Radii[J]. Journal of Geodesy and Geoinformation Science, 2020, 3(1): 45-51.

图/表 9

ΔR/a	Value of the maximum difference point	Difference maximum symbol expression
(R_a-R_e)/a	$π 2$	$23$ e²+ $512$ e⁴+ $13$ e⁶+ $55192$ e⁸
(R_a-R_F)/a	$π 2$	$23$ e²+ $1945$ e⁴+ $253756$ e⁶+ $32453113400$ e⁸
(R_a-R_V)/a	$π 2$	$23$ e²+ $49$ e⁴+ $115324$ e⁶+ $295972$ e⁸
(R_a-R_S)/a	$π 2$	$34$ e²+ $2764$ e⁴+ $85256$ e⁶+ $465516384$ e⁸

ΔR/a	Value of the minimum difference point	Difference minimum symbol expression
(R_a-R_e)/a	0	- $13$ e²- $112$ e⁴- $124$ e⁶- $5192$ e⁸
(R_a-R_F)/a	0	- $13$ e²- $790$ e⁴- $611512$ e⁶- $5969226800$ e⁸
(R_a-R_V)/a	0	- $13$ e²- $118$ e⁴- $13648$ e⁶- $353888$ e⁸
(R_a-R_S)/a	0	- $14$ e²- $564$ e⁴- $11256$ e⁶- $46516384$ e⁸

ΔR	Geodetic latitude B
R_a-R_e	$12$ arccos $13$ + $5224$ e²+ $24722304$ e⁴+ $231192331776$ e⁶+ $1076243221233664$ e⁸
R_a-R_F	$12$ arccos $13$ + $210$ e²+ $677212600$ e⁴+ $135132378000$ e⁶+ $9190336923492720000$ e⁸
R_a-R_V	$12$ arccos $13$ + $212$ e²+ $372864$ e⁴+ $9723456$ e⁶+ $3073121492992$ e⁸
R_a-R_S	$π 6$ + $3232$ e²+ $15723072$ e⁴+ $413212288$ e⁶+ $13633324718592$ e⁸

ΔR/a	Difference symbol expression
(R_e-R_V)/a	$136$ e⁴+ $7324$ e⁶+ $25615552$ e⁸
(R_F-R_V)/a	$145$ e⁴+ $231134$ e⁶+ $5891340200$ e⁸
(R_e-R_F)/a	$1180$ e⁴+ $1756$ e⁶- $251907200$ e⁸
(R_F-R_S)/a	$112$ e²- $12880$ e⁴- $12748384$ e⁶- $479119232243200$ e⁸
(R_e-R_S)/a	$112$ e²+ $1192$ e⁴- $1768$ e⁶- $11549152$ e⁸
(R_S-R_V)/a	- $112$ e²+ $13576$ e⁴+ $47520736$ e⁶+ $771553981312$ e⁸

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B	0°	15°	30°	45°	60°	75°	90°
R_a-R_e	-14256.5	-11404.6	-3599.9	7092.3	17820.5	25697.0	28584.9
R_a-R_F	-14254.9	-11403.0	-3598.4	7093.9	17822.1	25698.6	28586.4
R_a-R_S	-10696.8	-7844.9	-4036.8	10651.9	21380.1	29256.6	32144.5
R_a-R_V	-14248.5	-11396.6	-3592.0	7100.2	17828.5	25705.0	28592.8

Symbolic Expressions of Differences between Earth Radii

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