﻿ ﻿ ﻿ ﻿﻿ ﻿ 子午线弧长公式的简化及其泰勒级数解释
 文章快速检索 高级检索

1. 安徽农业大学 理学院，安徽 合肥 230036；
2. 江西省数字国土重点实验室，江西 抚州 344000

A Simplification of the Meridian Formula and Its Taylor-series Interpretation
GUO Jiachun1,2
1. School of Science,Anhui Agricultural University,Hefei 230036,China;
2. Jiangxi Province Key Lab for Digital Land,Fuzhou 344000,China
First author: GUO Jiachun(1981－),male,master of science,lecturer,majors in geodesy,cartography and GIS.E-mail： guojiachun@ahau.edu.cn
Abstract:A more concise formula of the meridian arc length was obtained by introduced two new parameters: the third flattening and the Gauss hypergeometric function. From another perspective,the simplified formula is also can be explained by a Taylor series expansion. By this,we got error estimate of the formula in terms of the Lagrange form of the remainder. For numerical verification of the error estimate theory,application example was presented by using the WGS84 data. The results show that experimental data are consistent with the error estimate theory and the simplified formula is more precise than the standard one.
Key words: meridian arc length     the third flattening     Gauss hypergeometric function     Taylor series     error estimate

1 引 言

2 子午线弧长公式的简化

3 子午线弧长公式的泰勒级数解释及其误差估计 3.1 公式的泰勒级数解释

3.2 误差估计

F(7)(ξ)情况类似，容易证明Me为收敛的幂级数。类似的，可得展开至其他各阶的误差估计式，结构与上述公式类似，此不赘述。

4 算例验证分析

 图 1 子午线弧长解算误差曲线图 Fig. 1 Error curves of the solution of the meridian arc length

 m 公式(1) 公式(10) 阶数 最大估计误差 实际最大绝对误差 阶数 最大估计误差 实际最大绝对误差 e6 1.78×10-2 1.35×10-2 n3 4.79×10-5 4.76×10-5 e8 1.32×10-4 8.97×10-5 n4 8.89×10-8 8.89×10-8 e10 1.00×10-6 5.97×10-7 n5 1.23×10-10 1.22×10-10 e12 7.69×10-9 3.97×10-9 n6 6.87×10-13 4.98×10-13 注：n4项的最大估计误差为8.886944×10-8m，实际最大误差为8.886899×10-8m，表中数据按四舍五入原则取位。

5 结 语

 [1] TORGE W. Geodesy[M]. 3rd ed. Berlin: Walter De Gruyter, 2001: 91-98. [2] KONG Xiangyuan, GUO Jiming, LIU Zongquan. Foundation of Geodesy[M]. Wuhan: Wuhan University Press, 2001: 64-73. (孔祥元,郭际明,刘宗泉. 大地测量学基础[M]. 武汉:武汉大学出版社,2001:64-73.) [3] HELMERT F R. Die Mathematischen und Physikalischen Theorien der Hheren Geodsie (English Translation Version) [M]. Leipzig: Printing and Publishing House of B.G. Teubner, 1880: 46-48. [4] DEAKIN R E, Hunter M N. Geometric Geodesy: Part A [R]. Melbourne: RMIT University, 2010: 60-77. [5] DING Jiabo. The Transforming the Zones of Gauss Projection from Latitudes of Low Points[J]. Acta Geodaetica et Cartographica Sinica, 1993, 22(3): 212-217. (丁佳波. 利用底点纬度进行高斯投影换代计算[J]. 测绘学报, 1993, 22(3): 212-217.) [6] LI Houpu, BIAN Shaofeng. The Expressions of Gauss Projection by Complex Numbers[J]. Acta Geodaetica et Cartographica Sinica, 2008, 37(1): 5-9. (李厚朴,边少峰. 高斯投影的复变函数表示[J]. 测绘学报,2008,37(1): 5-9.) [7] YANG Yuanxi. The Respective Roles and Contributions of Various Observation in Integrated Geodesy[J]. Acta Geodaetica et Cartographica Sinica, 1989,18(3):232-238. (杨元喜. 整体大地测量中各类观测值的分工与贡献[J]. 测绘学报,1989,18(3): 232-238.) [8] CHENG Pengfei, WEN Hanjiang, CHENG Yingyan, et al. Parameters of the CGCS 2000 Ellipsoid and Comparisons with GRS 80 and WGS 84[J]. Acta Geodaetica et Cartographica Sinica, 2009,38(3): 189-194. (程鹏飞,成英燕,文汉江, 等. 2000国家大地坐标系椭球参数与GRS 80和WGS 84的比较[J]. 测绘学报, 2009, 38(3): 189-194.) [9] LIU Zhengcai. Simplification of Formula of Meridian Arc Length & Program of Gauss Projection[J]. Engineering of Surveying and Mapping, 2001, 10(1):55-56. (刘正才. 子午线弧长公式的简化及通用高斯投影计算程序介绍[J]. 测绘工程, 2001, 10(1): 55-56.) [10] DORRER E. From Elliptic Arc Length to Gauss- Krüger Coordinates by Analytical Continuation [C]//Geodesy: The Challenge of the 3rd Millennium. Berlin: Springer, 2003: 293-298. [11] BERMEJO-SOLERA M, OTERO J. Simple and Highly Accurate Formulas for the Computation of Transverse Mercator Coordinates from Longitude and Isometric Latitude[J]. Journal of Geodesy, 2009, 83: 1-12. [12] GUO Jiachun, ZHAO Xiuxia, XU Li, etal. Calculating Meridian Arc Length by Transforming Its Formula into Elliptic Integral of Second Kind[J]. Journal of Geodesy and Geodynamics, 2011, 31(4): 94-98. (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.) [13] GUO Jiachun. New Method for Inverse Solution of Meridian Based on Elliptic Integral of the Second Kind[J]. Journal of Geodesy and Geodynamics, 2012, 32(3): 116-120. (过家春. 基于第二类椭圆积分的子午线弧长反解新方法[J]. 大地测量与地球动力学,2012,32(3): 116-120.) [14] 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. [15] YI Weiyong, BIAN Shaofeng, ZHU Hanquan. Determination of Foot Point Latitude by Analytic Positive Series[J]. Journal of Institute of Surveying and Mapping, 2000, 17(3): 167-171. (易维勇,边少峰,朱汉泉. 子午线弧长的解析型幂级数确定[J]. 测绘学院学报,2000,17(3): 167-171.) [16] LIU Renzhao, WU Jicang. Recursive Computation of Meridian Arc Length with Discretionary Precision[J]. Journal of Geodesy and Geodynamics, 2007, 27(5): 59-62. (刘仁钊,伍吉仓. 任意精度的子午线弧长递归计算[J]. 大地测量与地球动力学,2007,27(5): 59-62.) [17] BIAN S F, CHEN Y B. Solving an Inverse Problem of a Meridian Arc in Terms of Computer Algebra System[J]. Journal of Surveying Engineering, 2006, 132(1):7-10. [18] 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. (牛卓立. 以空间直角坐标系为参数的子午线弧长计算公式[J]. 测绘通报,2001(11): 14-15.) [19] LIU Xiushan. Numerical Integral Method Calculating Meridian Arc Length[J]. Bulletin of Surveying and Mapping, 2006(5): 4-6. (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.) [20] CHENG Pengfei, CHENG Yingyan, WEN Hanjiang, et al. Practical Manual on CGCS2000[M]. Beijing: Surveying and Mapping Press, 2008: 147-148. (程鹏飞,成英燕,文汉江, 等. 2000国家大地坐标系实用宝典[M]. 北京:测绘出版社,2008: 147-148.) [21] LIU Shishi, LIU Shida. Special Function[M]. Beijing: China Meteorological Press, 1988: 656-745. (刘式适,刘式达. 特殊函数[M]. 北京:气象出版社,1988:656-745.) [22] ZHU Huatong. Review on the Methods for Calculating Latitude of Low Points[J]. Bulletin of Surveying and Mapping, 1978(5): 10-14. (朱华统. 底点纬度计算方法评述[J]. 测绘通报,1978(5): 10-14.) [23] SUN Qun, YANG Qihe. The Research on the Computation of the Foot-point Latitude and the Inverse Solution of Isometric Latitude and Function[J].Journal of Institute of Surveying ang Mapping,1985(2):64-75. (孙群,杨启和. 底点纬度解算以及等量纬度和面积函数反解问题的探讨[J]. 测绘学院学报,1985(2): 64-75.) [24] WOLFRAM S. The Mathematica Book[M]. 5th ed. Champaign: Wolfram Media Inc, 2003. [25] BRONSHTEIN I N, SEMENDIAEY KA, HIRSCH K A. Handbook of Mathematics[M]. 4th ed. New York: Van Nostrand Reinhold, 2003: 412-416.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0017

0

#### 文章信息

GUO Jiachun.

A Simplification of the Meridian Formula and Its Taylor-series Interpretation

Acta Geodaetica et Cartographica Sinica,2014,43(2)：125-130.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0017