球面欧氏度量下Fermat-Torricelli点的问题

The Fermat-Torricelli Problem on Sphere with Euclidean Metric

研究球面上欧氏距离意义下Fermat-Torricelli点问题.给定边长分别为a, b, c的球面三角形△ABC,讨论当球面上点P到△ABC三个顶点A,B,C距离之和L达到最小时,求L,a,b,c之间满足的隐函数关系f(L,a,b,c)=0.将该问题转化成多元多项式方程组消元问题,结合Sylvester结式,Dixon结式,用符号数值混合计算方法进行隐函数插值,最终成功求出f(L,a,b,c),并说明对L,a,b,c之间可以满足的任意一个隐函数关系g(L,a, b, c)=0,g(L,a,b,c)均可用f(L,a,b,c)中4个不可约因子进行表示.

In this paper, we study the Fermat-Torricelli problem on sphere with Euclidean metric. Given a spherical triangle △ABC whose length of sides are a, b, c respectively, we discuss how to construct the implicit function f(L, a, b, c)= 0 when the sum of distances L between point P on sphere and the vertexes of △ABC reaches the minimum. We transform this problem to elimination of polynomial equations and successfully construct f(L,a,b, c) by combination of the Sylvester resultant, Dixon resultant and implicit function interpolation based on symbolic-numeric computation and then show that for any given g(L,a,b,c)= 0 which L,a,b,c may satisfy,g(L,a,b,c) can be expressed using the four irreducible factors of L(a, b, c).