-1

Suppose I have two spheres of arbitrary size $S_1$ and $S_2$ and I intersect them so they overlapped some extent.

What kind of shape would their intersection create? In other words, what does $S_1 \cap S_2$ look like?

EDIT:

Someone has suggested this answer which does seem relevant.

https://math.stackexchange.com/a/1551800/197705

When two sphere's $S_1$ and $S_2$ intersect, then their intersection forms a circle assuming they aren't tangent.

The answerer explains how to obtain the center and radius for this circle intersection of the two spheres.

However, to do this, they seem to rely on the distance from the center of the circle to each of the sphere centers, namely $d_1, d_2$.

How do we obtain $d_1$ and $d_2$? This step I don't get and so the answer seems circular to me.

Stan Shunpike
  • 4,921
  • @JoséCarlosSantos if indeed the intersection is a circle, then they is super helpful! – Stan Shunpike Feb 17 '24 at 21:49
  • Wait quick question, how do I know the orientation of that circle? The center doesn't provide that – Stan Shunpike Feb 17 '24 at 21:50
  • 1
    The circle lies on a plane orthogonal to the line defined by the centers of the spheres. – José Carlos Santos Feb 17 '24 at 21:52
  • @JoséCarlosSantos you were so helpful. Thank you very much sir! – Stan Shunpike Feb 17 '24 at 23:29
  • @JoséCarlosSantos but the post assumes $d_1$ and $d_2$ are given. Where would I get that information? How do I know how far the center of the circle is from the original spheres? – Stan Shunpike Feb 18 '24 at 00:13
  • @ronno How about this link? It particularly has "A useful trick: if f(x,y,z)=0 and g(x,y,z)=0 are the equations of two spheres $S_1$, and $S_2$ with distinct centers, then h=f−g is linear, and therefore h=0 is the equation of a plane P ... is contained in the third, ie equals the intersection of all three. Now, the intersection of two spheres is the intersection of a plane and a sphere" is a circle. https://math.stackexchange.com/questions/4860892/must-three-distinct-spheres-always-intersect-in-exactly-two-points/4860915#4860915 – nickalh Feb 18 '24 at 05:06
  • @JoséCarlosSantos The OP doubted the correctness of the answer in your linked question, and asked another question. – peterwhy Feb 20 '24 at 02:51
  • For clarity, I'm just asking questions. I have no idea what I doubt yet – Stan Shunpike Feb 20 '24 at 03:28

2 Answers2

1

It's a circle, as shown by this image:

enter image description here

Nate
  • 147
1

Note the calculation of $r$, the radius of the intersection circle, depends only on $d$, not on $d_1, d_2$. You can then use Pythagoras to find $d_1=\sqrt{r_1^2-r^2}, d_2=\sqrt{r_2^2-r^2}$

Ross Millikan
  • 374,822