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
周东权 1边少锋 2黄晓颖3
作者信息
- 1. 中国地质大学(武汉)地理与信息工程学院,湖北武汉 430078;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
- 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
- 折叠
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.
Abstract
The meridian arc length formula is usually expressed as a series expansion of the geodesic latitude B,whose coeffi-cients 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.
关键词
子午线弧长/第三扁率/数学分析/区域分析Key words
meridian arc length/third flattening/mathematical analysis/regional analysis引用本文复制引用
基金项目
国家自然科学基金(42342024)
中央高校基金(GLAB2024ZR05)
出版年
2024
NETL
NSTL
万方数据