Results 1 to 4 of 4

Math Help - Find center of sphere from 3 points

  1. #1
    Member
    Joined
    May 2008
    From
    Melbourne Australia
    Posts
    210
    Thanks
    25

    Find center of sphere from 3 points

    I know the radius R of a sphere and three points on its surface. I also know that the three points are below the horizontal plane through the center of the sphere.

    I need to find the equation of that sphere. Any ideas?

    My Attempt
    I can find the normal to the plane through A,B,C by using the cross product.

    Similarly I can find the angle between the vectors AB and AC using the dot product.

    With a bit of trig I think I can find the radius of the circle of intersection between the sphere and the plane. Assuming the angle between AB and AC is half of the angle between XB and XC where X is the center of the circle (not sure if this is right).

    From here I am a bit stumped.


    This problem comes from the rolling sphere method for determining lightning protection coverage.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2009
    Posts
    277
    Thanks
    2
    I think you've pretty much gotten it.

    My thinking is:

    3 points determine a circle.

    Once you have the circle, you can find its center (x) and radius (r). The plane of this circle is perpendicular to the line connecting the center of the sphere (X) to the center of the circle (x).

    Use the sphere radius (R) and (r) in a right triangle to find the distance
    between point x and the center of the sphere (X). R is the hypotenuse.

    There is an ambiguity between 2 possible spheres - one to the left of the circle and one to the right.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    From
    Melbourne Australia
    Posts
    210
    Thanks
    25
    I have not been able to find the centre of the circle. Only its radius and its normal.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2009
    Posts
    277
    Thanks
    2
    If we have 3 points on a circle, draw the segments connecting them. The intersection of the perpendicular bisectors of any 2 of those segments is the center of the circle.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: September 15th 2011, 11:21 AM
  2. Replies: 1
    Last Post: March 25th 2011, 02:49 PM
  3. Replies: 16
    Last Post: January 21st 2011, 12:37 PM
  4. Find Radius and Center of Sphere
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: October 12th 2008, 09:23 PM
  5. Speed of sphere center
    Posted in the Advanced Applied Math Forum
    Replies: 14
    Last Post: January 5th 2007, 10:02 AM

Search Tags


/mathhelpforum @mathhelpforum