基于第二类椭圆积分的椭圆弧长公式变换与应用  

Transformation Formulas for the Elliptic Arc Length and its Application based on the Elliptic Integral of the Second Kind

作　　者：过家春[1] 张庆国[1] 章林忠[1] 田劲松[1] 

机构地区：[1]安徽农业大学理学院,安徽合肥230036

出　　处：《数学的实践与认识》2011年第24期210-216,共7页Mathematics in Practice and Theory

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：以第二类椭圆积分为理论基础,通过推导,将椭圆弧长公式变换为以椭圆离心角、极角等常用角度参数为自变量的第二类椭圆积分的标准形式,建立起椭圆弧长公式与第二类椭圆积分标准形式之间的关系,并分析了椭圆上的弧微分变化规律及椭圆周长与离心率的变化关系.公式反映了椭圆弧长的本质问题即为第二类椭圆积分问题.因此,各类涉及椭圆弧长计算的应用问题,均可化为第二类椭圆的计算问题,应用时直接调用各类编程软件的函数库中的第二类椭圆积分函数,无需复杂编程即可实现椭圆弧长的高精度计算.文章以GPS采用的WGS-84椭球子午线弧长为例进行计算分析,验证了给出的公式及相关分析的正确性及应用价值.Based on the theory of the elliptic integral of the second kind, the elliptic arc length formula was transformed into the canonical form for the elliptic integral of the second kind by taking several angle parameters as the independent variables, such as the eccentric angle, polar angle etc., so that the function relations between them were established. Furthermore, the changing rule of the differential of the elliptic arc and the changing relationship between the elliptic perimeter and eccentricity were also analyzed. The formulas given in this paper got to the essence of the matter： various applications concerning the elliptic arc length all could be transformed into the solution of the elliptic integral of the second kind. Thus, the elliptic arc length could be easily calculated by calling ＂the elliptic integrals of the second kind＂ Function in various types of programming languages, which is more suitable for computer realization and the result has high precision. On these bases of theoretical analysis, the authors gave a practical example using the elliptic integrals of the second kind to calculate the meridian arc length of WGS-84 ellipsoid which is the reference ellipsoid used by the Global Positioning System （GPS）. The correctness and application values of the formulas and related analyses described in the paper were verified well by the example.

关 键 词：椭圆弧长 公式变换 第二类椭圆积分 弧微分 

分 类 号：O172.2[理学—数学]