Finding if two sphere's intersect method

[SanEng02]

Hi I was just curious if this method of solving whether or not two spheres intersect is a viable method that will give me the correct answer. Say if I am given the two equations of the sphere's is it viable to:

• Find the centre and radius of each sphere.
• Find the magnitude of the distance of the line between the sphere's centres
• If (magnitude distance of line) > radius they do not intersect, if (magnitude distance of the line) ≤ radius they do intersect.

From what I'm reading in the book and my notes, I think this should work.

Thanks in advance!

 

[Svein]

• If (magnitude distance of line) > radius they do not intersect, if (magnitude distance of the line) ≤ radius they do intersect.

Almost. Two points:

1. You assume that the spheres have the same radii. In the general case, they might not.
2. Assume that the spheres just barely touch. Then the distance from the center of each sphere to the touching point is equal to the radius of that sphere ∴ The distance between the centers is equal to the sum of the radii. Therefore, if the distance between the centers is less than the sum of the radii, the spheres will intersect.

 

[SanEng02]

Almost. Two points:

1. You assume that the spheres have the same radii. In the general case, they might not.
2. Assume that the spheres just barely touch. Then the distance from the center of each sphere to the touching point is equal to the radius of that sphere ∴ The distance between the centers is equal to the sum of the radii. Therefore, if the distance between the centers is less than the sum of the radii, the spheres will intersect.

In this case the radii were the same but I forgot to mention that but I knew I was missing something. Thanks!

 

[HallsofIvy]

Even in that case, your statement was wrong. Given two spheres of equal radii, they will intersect if and only if the distance between their centers is less than or equal to two times their common radius.

 

[SanEng02]

Right, that makes even more sense thanks!