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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.


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