测绘学报

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基于第二类椭圆积分的子午线弧长正反解新方法

过家春1,赵秀侠2,高飞   

  • 收稿日期:2011-03-02 修回日期:2011-07-06 出版日期:2011-12-28 发布日期:2019-01-01
  • 通讯作者: 过家春

New Method for Direct and Inverse Solutions of Meridian Based on the Elliptic Integral of the Second Kind

  • Received:2011-03-02 Revised:2011-07-06 Online:2011-12-28 Published:2019-01-01

摘要: 以第二类椭圆积分为理论基础,将子午线弧长正解公式变换为第二类椭圆积分的勒让德标准形式,对应的反解问题转换为第二类椭圆积分的求逆问题,理论上证明了计算子午线弧长的本质问题是第二类椭圆积分问题。在正算方面,将子午线弧长计算转换为第二类椭圆积分的计算,可以得到子午线弧长的真值(小数点后任意取位),而非已有算法的近似解,且计算效率得到显著提高。在反解方面,新方法为归化纬度余弦函数的常系数、解析型麦克劳林级数展开,收敛速度快,精度高,误差可以精确计算,实际应用时可根据精度需要展开至任意项。实例验算及分析表明,本文给出的正反解新方法具有简单、统一的数学模型,精度可靠,适用于不同的地球椭球,便于程序实现,可以推广使用。

Abstract: Based on the theory of the elliptic integral of the second kind, the meridian arc length formula was transformed into the Legendre’s canonical form for the elliptic integral of the second kind, so that the direct and inverse problems of Meridian could be solved by the elliptic integral of the second kind, and it revealed the essence of the problems. By the new method we described, the truth value of meridian arc length was calculated, and the computation efficiency was also further enhanced. On the other hand, an analytical method was offered to solve the inverse problem of Meridian. The new formula for inverse solution of Meridian was expressed by Maclaurin Series with constant coefficients that generated by the cosine function of reduced latitude. Numerical calculation indicated that the rate of convergence of the series is higher and the errors can be estimated accurately, thereby choosing the last term of the series easily in practical application. Theoretical analytics and application examples showed that the new method is more suitable for computer realization than the existing methods, and has a consolidated mathematical model suited for all earth ellipsoids.