Re: Point on sphere greatest distance from given points

*To*: mathgroup at smc.vnet.net*Subject*: [mg104076] Re: Point on sphere greatest distance from given points*From*: Ray Koopman <koopman at sfu.ca>*Date*: Sun, 18 Oct 2009 05:23:11 -0400 (EDT)*References*: <hbc81b$ckn$1@smc.vnet.net>

On Oct 17, 3:58 am, Kelly Jones <kelly.terry.jo... at gmail.com> wrote: > How can I use Mathematica to solve this problem: > > Given n points on a sphere, I want to find a point x such that: > > Sum[distance[x,i],{i,1,n}] > > is maximal, where "distance" is spherical ("great circle") distance. > > In other words, I want to find the point x "furthest" from the given n points. > > Is there any chance x will coincide with one of the given points? > If so, is there a better notion of distance to use? To avoid having the solution coincide with one of the given points, maximize the sum of the logs of the distances. s = Normalize/@RandomReal[NormalDistribution[0,1],{5,3}] {{-0.528071, -0.848197, 0.0412642}, {-0.0563032, -0.978864, -0.196608}, {0.750442, 0.305033, 0.586337}, {0.384831, 0.263578, 0.884552}, {0.922298, 0.0757753, 0.378977}} NMaximize[{Tr at Log[1-s.Normalize@{x,y,z}],x^2+y^2+z^2==1},{x,y,z}] {2.00971, {x -> -0.497972, y -> 0.585326, z -> -0.639858}} Although the final {x,y,z} is always normalized, the trial values are not, so we must normalize them manually.

**Follow-Ups**:**Re: Re: Point on sphere greatest distance from given points***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>