﻿ 常用地球半径差异符号表达式
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1. 海军工程大学导航工程系, 湖北 武汉 430033;
2. 32022部队, 湖北 武汉 430033

Symbolic expressions of differences between earth radius
ZONG Jingwen1 , LI Houpu1 , BIAN Shaofeng1 , TANG Qinghui2
1. Department of Navigation, Naval University of Engineering, Wuhan 430033, China;
2. 32022 Troops, Wuhan 430033, China
Foundation support: The National Natural Science Foundation of China (Nos. 41571441;41771487;41631072)
First author: ZONG Jingwen(1995—), male, PhD candidate, majors in geodetic surveying.E-mail:zjw19950613@163.com
Corresponding author: LI Houpu, E-mail: lihoupu1985@126.com
Abstract: A systematic and comprehensive comparison of the five commonly used earth radius in geodesy and cartography is carried out, and the differences between the most common points of the earth's radius, their corresponding maximum values, and the latitudes of equal points between themare derived with the help of computer algebraic systems. The symbolic expressionsare 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 degrees. The difference between the average radius of curvature and the equidistant 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 theoretical basis for relative research in earth science, space science, navigation and positioning.
Key words: earth radius     difference extrema value     extrema value point     symbolic expression     CGCS2000

1 常用地球半径定义

(1)

(2)

(3)

(4)

2 常用地球半径间的差异符号表达式

2.1 平均曲率半径与4种常用球体半径间的差异符号表达式

(5)

(6)

(7)

(8)

B取0时，存在最小值, 将其最小值进一步展开成偏心率e的幂级数形式

(9)

 ΔR/a 差异最大值点 差异最大值符号表达式 (Ra-Re)/a (Ra-RF)/a (Ra-RV)/a (Ra-RS)/a

 ΔR/a 差异最小值点 差异最小值符号表达式 (Ra-Re)/a 0 (Ra-RF)/a 0 (Ra-RV)/a 0 (Ra-RS)/a 0

(10)

(11)

(12)

 ΔR 大地纬度B Ra-Re Ra-RF Ra-RV Ra-RS

2.2 4种常用球体半径间的差异符号表达式

(13)

(14)

 ΔR/a 差值符号表达式 (Re-RV)/a (RF-RV)/a (Re-RF)/a (RF-RS)/a (Re-RS)/a (RS-RV)/a

3 算例分析

3.1 平均曲率半径与4种常用球体半径间的差异比较

 图 1 平均曲率半径与各常用地球半径的差异曲线图 Fig. 1 Chart of differences between average radius of curvature and four common sphere radii

 m B 0° 15° 30° 45° 60° 75° 90° Ra-Re -14 256.5 -11 404.6 -3 599.9 7 092.3 17 820.5 25 697.0 28 584.9 Ra-RF -14 254.9 -11 403.0 -3 598.4 7 093.9 17 822.1 25 698.6 28 586.4 Ra-RS -10 696.8 -7 844.9 -4 036.8 10 651.9 21 380.1 29 256.6 32 144.5 Ra-RV -14 248.5 -11 396.6 -3 592.0 7 100.2 17 828.5 25 705.0 28 592.8

 图 2 B∈[0°, 1°]内平均曲率半径与各常用地球半径的差异曲线图 Fig. 2 Chart of differences between average radius of curvature and four common sphere radii

 大地纬度 Ra=Re Ra=RF Ra=RS Ra=RV B 35°19′31.77″ 35°19′7.79″ 30°3′45.04″ 35°19′35.11″

3.2 4种常用球体半径间的差异比较

 m Re-RV Re-RS Re-RF RF-RS RF-RV RS-RV ΔR 7.98 3 559.63 1.59 3 558.04 6.39 -3 551.64

4 结论

(1) 平均曲率半径与4种常用球体半径差异值随大地纬度的增大而增大，差异绝对值随大地纬度的增大先减小后增大，在某一个特定点处差异值为0。其中，当B时，存在差异最大值，平均曲率半径与等距离球半径间差异最大，差异最大符号表达式的首项为B取0时，存在差异最小值，平均曲率半径与平均球半径间差异值最小，最小值符号表达式的首项为

(2) 平均曲率半径是微观上的平均值，是随着大地纬度B逐点变化的；4种常用球体半径是宏观意义上的地球球体上的平均值，与大地纬度B无关。平均曲率半径与4种常用球体半径除在特殊点等价外，其余点无法进行相互代替应用。

(3) 4种常用球体半径间，平均球半径和等面积半径间的差异绝对值最小，绝对值最小符号表达式的首项为；平均球半径和等距离半径之间的差异绝对值最大，绝对值最大值符号表达式的首项为。等距离球半径与其余3种地球半径差异较大，不能相互替代；平均球半径和等体积球半径与等面积球半径之间差异较小，在实际应用过程中可以用平均球半径和等体积球半径来替代等面积球半径。

(4) 将常用地球半径间差异值表示为符号形式，并统一展开为偏心率e的幂级数形式，该表达式易于比较分析，一定程度上丰富了测量及地图学数学分析理论。

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http://dx.doi.org/10.11947/j.AGCS.2019.20180145

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#### 文章信息

ZONG Jingwen, LI Houpu, BIAN Shaofeng, TANG Qinghui

Symbolic expressions of differences between earth radius

Acta Geodaetica et Cartographica Sinica, 2019, 48(2): 238-244
http://dx.doi.org/10.11947/j.AGCS.2019.20180145