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摘要: 为了解决目前贝塞尔大地主题反解算法不统一与存在适用条件限制的问题, 根据球面三角形正、余弦定理和拉格朗日级数理论, 用计算机代数系统推导并给出一种改进的大地主题直接反解的微分改正方法。该算法消除了逐次趋近计算, 适用传统反解算法中的奇异情况, 解决了求解三角方程中方位角的多值对应问题。试验结果表明, 当大地线长度小于20000km时, 算法精度高达0.0001s, 具有通用性, 特别适用于电算化, 对远洋大地线航法计算具有一定的应用价值。Abstract: In order solve the nonuniform algorithms of Bessel's inverse problem and the application restrictions of the algorithms, sine and cosine theories based on sphere triangle and Lagrange series theory were analyzed, and an improved arithmetic of differential correction for the direct inverse solution of geodetic problem was put forward by using computer algebra system. Simulation result shows that the improved algorithm is suitable for the arbitrarily special situations without iterative calculations, and has high accuracy of 0.000 1 s when geodetic length is less than 20 000 km. The azimuth multiple-valued corresponding problem in the solution of trigonometric equation is solved. The improved algorithm is also suitable for the program implementation without generality loss, and can be applied in the area of ocean geodetic line sailing computation.
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表 1 长距离大地主题反算结果
Table 1. Inverse solution of geodetic problem on long distance
参数类型 算例1 a 6 378 388.0 m f 1.0/297.0 L1 010°00′00.″000 0 E B1 50°00′00.″000 0 N H1 0 m L2 105°05′38.″2 994 E B2 -62°57′03.″2 038 N H2 0 m s 14 999.999 996 457 848 km A1 140°00′00.″000 000 A2 294°46′41.″484 147 
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表 2 起点在最高纬度点处的大地主题反算结果
Table 2. Inverse solution of geodetic problem starting at the highest latitude
参数类型 算例2 a 6 378 245.0 m f 1.0/298.3 L1 35°00′00.″000 000 E B1 35°00′00.″000 000 N H1 0 m L2 173°13′02.″584 287 E B2 27°42′53.″569 421 N H2 0 m s 16 000.000 001 km A1 089°59′59.″999 859 A2 292°13′54.″917 935
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表 3 终点在最高纬度点处的大地主题反算结果
Table 3. Inverse solution of geodetic problem ending at the highest latitude
参数类型 算例3 a 6 378 245.0 m f 1.0/298.3 L1 043°52′51.″265 641 E B1 24°00′11.″828 134 S H1 0 m L2 173°00′00.″000 000 E B2 35°00′00.″000 000 N H2 0 m s 15 000.000 000 km A1 063°47′27.″170 549 A2 270°00′00.″000 012
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表 4 起点在南纬终点在北极的大地主题反算结果
Table 4. Inverse solution of geodetic problem starting at south latitude and ending at north pole
参数类型 算例4 a 6 378 245.0 m f 1.0/298.3 L1 070°00′00.″000 000 E B1 15°00′00.″000 000 S H1 0 m L2 070°00′00.″000 000 E B2 90°00′00.″000 000 N H2 0 m s 11 661.156 725 km A1 000°00′00.″000 000 A2 180°00′00.″000 000
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