What shape does the intersection of two spheres yield? How do you find the center and radius of this circle?

Question

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?

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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.

Answer 1

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

Answer 2

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 $d_1=\sqrt{r_1^2-r^2}, d_2=\sqrt{r_2^2-r^2}$