常用海图投影平面上大椭圆航线的表象与曲率分析

Representation and curvature analysis of great ellipse on common chart projection plane

利用空间矢量方法推导出了椭球面上只与起止点地理坐标有关的大椭圆航线方程,代入4种常用海图投影的正解公式,得到不同投影平面上的大椭圆参数方程;利用上述参数方程进而推导出了不同投影面上大椭圆航线的曲率与曲率半径公式。选取伦敦到纽约的大椭圆航线为例,通过绘制不同投影面上的大椭圆航线并分析其曲率、曲率半径变化曲线可知,大椭圆航线在日晷投影上的表象为曲率处处为0的直线,而在其他投影面上的表象为曲率较小但不断变化的曲线。利用推导的曲率半径公式可以计算各类大椭圆航线上任意位置的“代曲直距”,方便在不同比例尺的海图上对大椭圆航线进行量测和绘制,提高作图效率。

By using the space vector method, the equation of great ellipse on ellipsoidal surface only related to the geographical coordinates of the starting and ending points was derived, the parameter equations of great ellipse routes on various projection planes were obtained from the great ellipse equation and the positive solution formulas of the four kinds of common projection. And then the curvature and radius of curvature of great ellipse routes on the four kinds of projection planes were derived. The great ellipse route from London to New York was taken as an example, by drawing the great ellipse route on different projection planes and analyzing the curve of curvature and curvature radius, the conclusion is drawn that the representation of great ellipse route on gnomonic projection plane is straight line, while which of great ellipse route on the other projection planes are curves with curvature changing slightly. The formulas of curvature radius derived in this paper can be used to calculate the “substitution distance” of great ellipse routes, which is convenient to measure and draw the great ellipse routes on the nautical chart with any scale, and improve the efficiency of nautical drawing.