子午线弧长正解展开通式及数学分析

doi:10.11947/j.AGCS.2024.20230546

测绘学报 ›› 2024, Vol. 53 ›› Issue (9): 1790-1798.doi: 10.11947/j.AGCS.2024.20230546

• 大地测量与导航 • 上一篇

子午线弧长正解展开通式及数学分析

周东权¹^,²(), 边少锋²(), 黄晓颖³

^1.中国地质大学（武汉）地理与信息工程学院，湖北　武汉　430078
^2.中国地质大学（武汉）地质探测与评估教育部重点实验室，湖北　武汉　430074
^3.海军工程大学作战运筹与规划系，湖北　武汉　430033

收稿日期:2023-12-05 发布日期:2024-10-16
通讯作者: 边少锋 E-mail:1431645283@qq.com;sfbian@sina.com
作者简介:周东权（1999—），男，硕士，研究方向为椭球大地测量。E-mail：1431645283@qq.com
基金资助:
国家自然科学基金(42342024);中央高校基金(GLAB2024ZR05)

General series expansion and mathematical analysis of the direct solution of meridian arc length

Dongquan ZHOU¹^,²(), Shaofeng BIAN²(), Xiaoying HUANG³

^1.School of Geography and Information Engineering, China University of Geosciences (Wuhan), Wuhan 430078, China
^2.Key Laboratory of Geological Survey and Evaluation of Ministry of Education, China University of Geosciences (Wuhan), Wuhan 430074, China
^3.Department of Operational Research and Programing, Naval University of Engineering, Wuhan 430033, China

Received:2023-12-05 Published:2024-10-16
Contact: Shaofeng BIAN E-mail:1431645283@qq.com;sfbian@sina.com
About author:ZHOU Dongquan (1999—), male, master, majors in ellipsoidal geodesy. E-mail: 1431645283@qq.com
Supported by:
The National Natural Science Foundation of China(42342024);The Opening fund of Key Laboratory of Geological Survey and Evaluation of Ministry of Education(GLAB2024ZR05)

摘要/Abstract

摘要：

子午线弧长公式通常表示为大地纬度B的级数展开式，其系数以第一偏心率e为参数，本文利用第三扁率n对公式进行重新推导，并将其表示为三角函数的3种形式：倍角形式、指数形式和二倍角形式。重新推导的公式中各系数中的分母值都明显变小，个别高阶项系数消失，结构简单、形式简洁。基于此，对3种表示形式的8阶展开式和10阶展开式进行截断误差分析，结果表明3种表示形式的8阶展开式精度最低也有毫米级，能满足日常的使用场景，而10阶展开式精度提高了至少2个数量级，能满足高精度的使用场景。在高精度的使用场景下，不同表示形式的子午线弧长公式具有不同的实用性，分析表明在0°N—30°N低纬度地区推荐使用三角函数的指数形式，在30°N—55°N中纬度地区推荐使用三角函数的二倍角形式，而在55°N—90°N高纬度地区推荐使用三角函数的倍角形式。

关键词: 子午线弧长, 第三扁率, 数学分析, 区域分析

Abstract:

The meridian arc length formula is usually expressed as a series expansion of the geodesic latitude B, whose coefficients are parameterized by first eccentricity e. In this paper, the formula is rederived using third flattening n and expressed as three forms of trigonometric functions: multiple angle form, exponential form, and double dangle form.The denominator value in each coefficient in the rederived formula is obviously smaller, and the individual higher-order term coefficients disappear, with a simple structure and concise form. Based on this, the truncation error analysis of the 8th-order and 10th-order expanded formulas of the three representations shows that the accuracy of the 8th-order expanded formulas of the three representations is at least millimeters, which can satisfy the daily use scenarios, while the accuracy of the 10th-order expanded formulas has been improved by at least two orders of magnitude, which can satisfy the high-precision use scenarios. Under the high-precision use scenario, the meridian arc length formulas of different representations have different practicality, and the analysis shows that the exponential form of trigonometric function is recommended at the low latitude of 0°N—30°N, double angle form of trigonometric function is recommended at the middle latitude of 30°N—55°N, and the multiple angle form of trigonometric function is recommended at the high latitude of 55°N—90°N.

Key words: meridian arc length, third flattening, mathematical analysis, regional analysis

中图分类号:

P223

周东权, 边少锋, 黄晓颖. 子午线弧长正解展开通式及数学分析[J]. 测绘学报, 2024, 53(9): 1790-1798.

Dongquan ZHOU, Shaofeng BIAN, Xiaoying HUANG. General series expansion and mathematical analysis of the direct solution of meridian arc length[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(9): 1790-1798.

图/表 9

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参考文献 28

[1]	孔祥元, 郭际明, 刘宗泉. 大地测量学基础[M]. 2版. 武汉: 武汉大学出版社, 2001.
	KONG Xiangyuan, GUO Jiming, LIU Zongquan. Foundation of geodesy[M]. 2nd version. Wuhan: Wuhan University Press, 2001.
[2]	王会, 王晓栋. 常微分方程数值解法在子午线正反算中的应用[J]. 铁道勘察, 2018, 44(1):15-18.
	WANG Hui, WANG Xiaodong. Application of numerical solution of ordinary differential equation in the positive and negative calculation of meridian[J]. Railway Investigation and Surveying, 2018, 44(1):15-18.
[3]	杨双富. 再议计算子午线弧长的数值积分法[J]. 城市勘测, 2010(6):140-142, 145.
	YANG Shuangfu. Reconsideration of the numerical calculation of radial arc length integral method[J]. Urban Geotechnical Investigation & Surveying, 2010(6):140-142,145.
[4]	熊介. 椭球大地测量学[M]. 北京: 解放军出版社, 1988.
	XIONG Jie. Ellipsoidal geodesy[M]. Beijing: PLA Press, 1988.
[5]	郑红晓, 张红方, 雷伟伟. 子午线弧长计算的数值积分算法及其比较[J]. 铁道勘察, 2014, 40(6):8-10.
	ZHENG Hongxiao, ZHANG Hongfang, LEI Weiwei. The calculation and comparison of meridian arc length based on numerical integral algorithm[J]. Railway Investigation and Surveying, 2014, 40(6):8-10.
[6]	TSENG W K, CHANG Weijie, PEN C L. New meridian arc formulae by the least squares method[J]. Journal of Navigation, 2014, 67(3):495-510.
[7]	TSENG W K, TSAI K C, LIOU Chian, et al. Algorithms for the generalized inverse solution and direct solution: using an algebra computer-based system to obtain meridian arc length[J]. Journal of Marine Science and Technology, 2023, 31(2):9.
[8]	边少锋, 李厚朴. 大地测量计算机代数分析[M]. 北京: 科学出版社, 2018.
	BIAN Shaofeng, LI Houpu. Computer algebra analysis on geodesy[M]. Beijing: Science Press, 2018.
[9]	刘学杰, 杨丽坤. 子午线弧长的计算方法及精度分析[J]. 测绘通报, 2017(8):106-109, 116.
	LIU Xuejie, YANG Likun. Calculation methods and accuracy analysis of meridian arc length[J]. Bulletin of Surveying and Mapping, 2017(8):106-109, 116.
[10]	王元波, 杨丽坤, 张洁. 计算子午线弧长的三类算法及其分析比较[J]. 测绘与空间地理信息, 2017, 40(9):99-102.
	WANG Yuanbo, YANG Likun, ZHANG Jie. The three kinds of algorithm for calculating meridian arc length and their analysis and comparison[J]. Geomatics & Spatial Information Technology, 2017, 40(9):99-102.
[11]	杨丽坤, 雷伟伟. 计算子午线弧长与底点纬度的常微分方程数值解法[J]. 测绘科学技术学报, 2017, 34(6):560-563.
	YANG Likun, LEI Weiwei. Calculation of meridian arc length and latitude of pedal based on the numerical solution of ordinary differential equations[J]. Journal of Geomatics Science and Technology, 2017, 34(6):560-563.
[12]	李松林, 李厚朴, 边少锋, 等. 等角纬度与子午线弧长变换的直接表达式[J]. 海洋测绘, 2019, 39(2):21-25.
	LI Songlin, LI Houpu, BIAN Shaofeng, et al. Direct expansions of transformation between conformal latitude and meridian arc length[J]. Hydrographic Surveying and Charting, 2019, 39(2):21-25.
[13]	刘修善. 计算子午线弧长的数值积分法[J]. 测绘通报, 2006(5):4-6.
	LIU Xiushan. Numerical integral method of calculating meridian arc length[J]. Bulletin of Surveying and Mapping, 2006(5):4-6.
[14]	牛卓立. 以空间直角坐标为参数的子午线弧长计算公式[J]. 测绘通报, 2001(11):14-15.
	NIU Zhuoli. Formulae for calculation of meridian arc length by the parameters of space rectangular coordinates[J]. Bulletin of Surveying and Mapping, 2001(11):14-15.
[15]	边少锋, 李厚朴, 李忠美. 地图投影计算机代数分析研究进展[J]. 测绘学报, 2017, 46(10):1557-1569. DOI: 10.11947/j.AGCS.2017.20170396.
	BIAN Shaofeng, LI Houpu, LI Zhongmei. Research progress in mathematical analysis of map projection by computer algebra[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(10):1557-1569. DOI: 10.11947/j.AGCS.2017.20170396.
[16]	李厚朴, 边少锋, 钟斌. 地理坐标系计算机代数精密分析理论[M]. 北京: 国防工业出版社, 2015.
	LI Houpu, BIAN Shaofeng, ZHONG Bin. Precise analysis theory of computer algebra in geographical coordinate system[M]. Beijing: National Defense Industry Press, 2015.
[17]	RATH G, BESSEL R. Bestimmung der axen des elliptischen rotationssphäroids, welches den vorhandenen messungen von meridianbögen der erde am meisten entspricht[J]. Astronomische Nachrichten, 1837, 14(23):333-346.
[18]	李晓勇, 李厚朴, 刘国辉, 等. 等面积纬度函数与常用纬度间的直接变换[J]. 海洋测绘, 2022, 42(2):78-82.
	LI Xiaoyong, LI Houpu, LIU Guohui, et al. Direct transformation between equal-area latitude function and common latitude[J]. Hydrographic Surveying and Charting, 2022, 42(2):78-82.
[19]	HELMERT F R. Die mathematischen und physikalischen theorieen der höheren geodäsie[M]. Leipzig: Druck und Verlag von B.G. Teubner, 1884: 46-48.
[20]	BIAN Shaofeng, CHEN Yongbing. Solving an inverse problem of a meridian arc in terms of computer algebra system[J]. Journal of Surveying Engineering, 2006, 132(1):7-10.
[21]	BOWRING B R. New equations for meridional distance[J]. Bulletin Géodésique, 1983, 57(1):374-381.
[22]	KARNEY C F F. On auxiliary latitudes[J]. Survey Review, 2024, 56(395):165-180.
[23]	KAWASE K. A general formula for calculating meridian arc length and its application to coordinate conversion in the Gauss-Krüger projection[J]. Bulletin of the Geospatial Information Authority of Japan, 2011, 59:1-13.
[24]	过家春, 赵秀侠, 徐丽, 等. 基于第二类椭圆积分的子午线弧长公式变换及解算[J]. 大地测量与地球动力学, 2011, 31(4):94-98.
	GUO Jiachun, ZHAO Xiuxia, XU Li, et al. Calculating meridian arc length by transforming its formulae into elliptic integral of second kind[J]. Journal of Geodesy and Geodynamics, 2011, 31(4):94-98.
[25]	过家春, 李厚朴, 庄云玲, 等. 依不同纬度变量的子午线弧长正反解公式的级数展开[J]. 测绘学报, 2016, 45(5):560-565. DOI: 10.11947/j.AGCS.2016.20140575.
	GUO Jiachun, LI Houpu, ZHUANG Yunling, et al. Series expansion for direct and inverse solutions of meridian in terms of different latitude variables[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(5):560-565. DOI: 10.11947/j.AGCS.2016.20140575.
[26]	过家春. 子午线弧长公式的简化及其泰勒级数解释[J]. 测绘学报, 2014, 43(2):125-130. DOI: 10.13485/j.cnki.11G2089.2014.0017.
	GUO Jiachun. A simplification of the meridian formula and its Taylor-series interpretation[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(2):125-130. DOI: 10.13485/j.cnki.11G2089.2014.0017.
[27]	汪绍航, 边少锋, 金立新, 等. 椭球大地测量常用幂级数的第三扁率展开[J]. 测绘科学技术学报, 2021, 38(6):571-578.
	WANG Shaohang, BIAN Shaofeng, JIN Lixin, et al. Third flattening expansion of common power series in ellipsoidal geodesy[J]. Journal of Geomatics Science and Technology, 2021, 38(6):571-578.
[28]	周东权, 刘敏, 魏冲, 等. 常用投影大地线的高效展绘及Mathematica实现[J]. 现代导航, 2023, 14(6):422-430.
	ZHOU Dongquan, LIU Min, WEI Chong, et al. Efficient mapping of common projected geodetic lines and mathematica implementation[J]. Modern Navigation, 2023, 14(6):422-430.

表示形式	8阶截断误差计算通式	10阶截断误差计算通式
三角函数的倍角形式	ε₁₁≈a(1－n)b₁₀sin(10B)	ε₁₂≈a(1－n)b₁₂sin(12B)
三角函数的指数形式	ε₂₁≈a(1－n)cos B×c₉sin⁹B	ε₂₂≈a(1－n)cos B×c₁₁sin¹¹B
三角函数的二倍角形式	ε₃₁≈a(1－n)sin(2B)×d₅cos⁴(2B)	ε₃₂≈a(1－n)sin(2B)×d₆cos⁵(2B)

子午线弧长正解展开通式及数学分析

General series expansion and mathematical analysis of the direct solution of meridian arc length

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表示形式	阶数	极值/m
三角函数的倍角形式	8阶	±4.244 18×10^－7
三角函数的倍角形式	10阶	±6.977 73×10^－11
三角函数的指数形式	8阶	－0.000 046 386
三角函数的指数形式	10阶	2.556 33×10^－8
三角函数的二倍角形式	8阶	±2.107 85×10^－8
三角函数的二倍角形式	10阶	±5.778 77×10^－10

[1]	过家春, 李厚朴, 庄云玲, 李大军, 吴艳兰. 依不同纬度变量的子午线弧长正反解公式的级数展开[J]. 测绘学报, 2016, 45(5): 560-565.
[2]	过家春. 子午线弧长公式的简化及其泰勒级数解释[J]. 测绘学报, 2014, 43(2): 125-130.