单位球面上加权Fermat-Torricelli问题的解析解
An analytical solution of the weighted Fermat-Torricelli problem on the unit sphere
摘要：
对于由三条长度为Pi/2的相等测地弧（边）组成的等边测地三角形A_1A_2A_3，对于给定的三个正不等权对应于单位球面上的三个顶点，我们得到了加权Fermat-Torricelli问题的一个解析解。这个解析解是文[4]中Cockayne解的推广。进而，应用几何塑性原理和球余弦定理，导出了加权Fermat-Torricelli点的一个必要条件，其形式为关于测地弧A_1A_1 '的长度的三个超越方程，A_2A_2 '和A_3A_3'，将加权Fermat-Torricelli点A_0定位在边长小于Pi/2的单位球面上的测地线三角形A_1 'A_2' A_3 '的内部。
英文摘要：
We obtain an analytical solution for the weighted Fermat-Torricelli problem for an equilateral geodesic triangle A_1A_2A_3 which is composed by three equal geodesic arcs (sides) of length Pi/2 for given three positive unequal weights that correspond to the three vertices on a unit sphere. This analytical solution is a generalization of Cockayne's solution given in [4] for three equal weights. Furthermore, by applying the geometric plasticity principle and the spherical cosine law, we derive a necessary condition for the weighted Fermat-Torricelli point in the form of three transcedental equations with respect to the length of the geodesic arcs A_1A_1', A_2A_2'and A_3A_3'to locate the weighted Fermat-Torricelli point A_0 at the interior of a geodesic triangle A_1'A_2'A_3'on a unit sphere with sides less than Pi/2.
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