General series expansion and mathematical analysis of the direct solution of meridian arc length
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.
meridian arc lengththird flatteningmathematical analysisregional analysis
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