基于第二类椭圆积分的子午线弧长公式变换及解算  

CALCULATING MERIDIAN ARC LENGTH BY TRANSFORMING ITS FORMULAE INTO ELLIPTIC INTEGRAL OF SECOND KIND

作　　者：过家春[1] 赵秀侠[2] 徐丽[1] 田劲松[1] 高飞[3] 

机构地区：[1]安徽农业大学理学院,合肥230036 [2]安徽大学资源与环境工程学院,合肥230039 [3]合肥工业大学土木与水利工程学院,合肥230009

出　　处：《大地测量与地球动力学》2011年第4期94-98,共5页Journal of Geodesy and Geodynamics

基　　金：国家自然科学基金(40771117);国家农业信息化工程技术研究中心开放课题(KF2010W40-046)

摘　　要：通过推导,将子午线弧长公式变换为基于第二类椭圆积分的两种形式:"形式Ⅰ"将子午线弧长公式表达为一个有理函数和第二类椭圆积分之和,建立了以大地纬度B为自变量的子午线弧长公式与第二类椭圆积分之间的关系;"形式Ⅱ"给出了以归化纬度μ为自变量、直接利用第二类椭圆积分计算子午线弧长的公式。利用此两种形式的子午线弧长公式,在Matlab中编写程序,调用第二类椭圆积分函数EllipticE(x,k)计算子午线弧长,精度和计算效率均优于经典算法。对CGCS2000所采用的地球椭球子午线弧长的计算表明,此两种形式的子午线弧长公式建立了子午线弧长公式与第二类椭圆积分的关系,结构简洁,易于展开,一定程度上完善了子午线弧长理论,且便于手工计算及计算机程序实现。Abstract A new idea that by transforming the meridian arc length formula into two other forms was put forwand, which are expressed by the elliptic integrals of the second kind, they were named as ＂Form I ＂ and ＂Form Ⅱ ＂ In ＂Form I ＂, the meridian arc length formula is represented as the sum of a rational function and the elliptic integrals of the second kind by the geodetic latitude parameter, which establishs the function relations between the meridian arc length and the elliptic integrals of the second kind. Analogously, taking the reduced latitude as an in dependent variable, the ＂Form Ⅱ ＂ formula give a simpler form of the meridian arc length formula in terms of the elliptic integrals of the second kind. On these bases of theoretical analysis, the computer program is compiled in MATLAB by calling the Elliptic E （x, k） Function to calculate the meridian arc length. It was proved that this method improved greatly the accuracy and efficiency of previous calculation. Moreover, the meridian arc length of CGCS2000 was also calculated based on the principle that provided. The results indicate that the ＂Form I ＂ and＂FormⅡ ＂ formula are simpler and more suitable for series expansions and the realization on computer than the classical form. Furthermore, it perfects the meridian theory.

关 键 词：子午线弧长 公式变换 椭圆积分 大地纬度 归化纬度 

分 类 号：P226[天文地球—大地测量学与测量工程]