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

doi:10.11947/j.AGCS.2024.20230546

Abstract

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

CLC Number:

P223

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.

Figures/Tables 9

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References 28

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表示形式	正解展开通式
三角函数的倍角形式
三角函数的指数形式
三角函数的二倍角形式

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)

表示形式	截断误差计算通式
三角函数的倍角形式
三角函数的指数形式
三角函数的二倍角形式

表示形式	8阶截断误差极值点通式	10阶截断误差极值点通式
三角函数的倍角形式
三角函数的指数形式
三角函数的二倍角形式

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