﻿ 常用海图投影平面上大椭圆航线的表象与曲率分析
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Representation and curvature analysis of great ellipse on common chart projection plane
LI Songlin, CHEN Cheng, BIAN Shaofeng, LI Houpu, LIU Qiang
Department of Navigation, Naval University of Engineering, Wuhan 430033, China
Abstract: 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.
Key words: great ellipse route    nautical chart projection    curvature    radius of curvature    substitution distance

1 常用投影平面上的大椭圆航线参数方程

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 图 1 大椭圆航线示意图 Fig. 1 Sketch of great ellipse route

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2 常用投影平面上大椭圆航线的曲率与曲率半径公式推导

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2.1 墨卡托投影平面上大椭圆航线曲率公式

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2.2 高斯投影平面上大椭圆航线曲率公式

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2.3 极球面投影平面上大椭圆航线曲率公式

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2.4 日晷投影平面上大椭圆航线曲率公式

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3 算例分析

 图 2 常用投影平面上大椭圆航线曲率变化曲线 Fig. 2 Curves of curvature of great ellipse routes on common projection planes

 图 3 常用投影平面上大椭圆航线曲率半径变化曲线 Fig. 3 Curves of curvature radius of great ellipse routes on common projection planes

 图 4 常用投影平面上的大椭圆航线 Fig. 4 Great ellipse route on common projection plane

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 投影方式 经度l 曲率绝对值极大值κmax/m-1 曲率半径极小值Mmin/m 比例尺 最大代曲直距Smax/cm 墨卡托投影 -23°1.5′ 1.262 4×10-7 7.921 4×106 1：500 000 11.258 高斯投影 -74° 5.342 7×10-8 1.871 7×107 1：500 000 17.305 极球面投影 -74° 4.664 6×10-8 2.143 9×107 1：500 000 18.520 日晷投影 — 0 ∞ — ∞

4 结论

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http://dx.doi.org/10.11947/j.AGCS.2019.20180348

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#### 文章信息

LI Songlin, CHEN Cheng, BIAN Shaofeng, LI Houpu, LIU Qiang

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

Acta Geodaetica et Cartographica Sinica, 2019, 48(10): 1331-1338
http://dx.doi.org/10.11947/j.AGCS.2019.20180348