Thread: Finding point on sphere furthest from point in space

Find the point on the sphere which is furthest from the point C(1,3,4).

Now I know that you suppose to find an equation of a line relating the centre of the sphere (2,2,7) and C(1,3,4) e.g. x = (1,3,4) + t(1,-1,3) where t can be any real number. But I dont understand what you would do to find the furthest point on the sphere?

Originally Posted by SyNtHeSiS

Find the point on the sphere which is furthest from the point C(1,3,4).

Now I know that you suppose to find an equation of a line relating the centre of the sphere (2,2,7) and C(1,3,4) e.g. x = (1,3,4) + t(1,-1,3) where t can be any real number. But I dont understand what you would do to find the furthest point on the sphere?

Think about it geometrically. To get from C to the furthest point on the sphere, you have to go from C to the centre of the sphere and then continue in the same direction for a distance equal to the radius of the sphere, which is . You have already identified the direction of travel as the vector (1,–1,3), which has length .

Those s can't be coincidental, can they?

Hello, SyNtHeSiS!

Code:

              * * *         o C
          *           *   *
        *               ♥ P
       *              *  *
                    *
      *           *       *
      *         o         *
      *       * A         *
            *
       *  *              *
      Q ♥               *
      *   *           *
    *         * * *

Construct a line through and the center of the sphere

It will intersect the sphere in two points, and
. . We want , the point further from

The line from to has direction

The line has parametric equations: .


Substitute into the equation of the sphere:

. .

. .

. .





I suspect that point is further from , but let's make sure.


Distance from to

. .


Distance from to

. .

Hence, is the point we are seeking.


Therefore: .

Originally Posted by Soroban

The line from to has direction

Thanks for the great post. I was just curious why is it that choosing a line from A(2,2,7) to C(1,3,4) with direction <-1,1,-3> equivalent to the quote above? I thought this vector be the same magnitude but opposite, since points A and C is switched around.

Just to point out that the makings of a much simpler solution were already in the original post, where the equation of the line from C to the centre of the sphere is given as . If you put t=1 in that equation then you get the point (2,2,7) (the centre of the sphere). To get to the furthest point of the sphere you have to travel a further distance along that same line. The extra distance that you need to travel is units, which is three times the length of the vector (1,–1,3).

So to get to the furthest point of the sphere you need to take t = 4 (one unit of t to take you to the centre of the sphere, and another three units of t to take you to the furthest point). The furthest point is therefore at (1,3,4) + 4(1,–1,3) = (5,–1,16).