# Thread: Find center of sphere from 3 points

1. ## 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.

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.

3. I have not been able to find the centre of the circle. Only its radius and its normal.

4. 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.