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If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of intersection between two spheres?

Kay K.
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Ben-ZT
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1 Answers1

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Assume that you also know the distance between centers of those two spheres. Let's say it's $d$. And the radius of the common circle is $r$. The distance from the center of the new circle to two spheres are $d_1$ and $d_2$. Then,

$$d_1^2+d_2^2=(r_1^2-r^2)+(r_2^2-r^2)=d^2$$ $$\therefore r=\sqrt{\frac{r_1^2+r_2^2-d^2}{2}}$$

And then the center ($C$) of the new circle is an interpolation of centers of two spheres ($P_1$ and $P_2$). $$C=\frac{d_2P_1+d_1P_2}{d_1+d_2}$$

Kay K.
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    For $r_1 = r_2 = d = 1$, the radius of the common circle $r$ should be the height of an equilateral triangle of unit side: $r=\frac{\sqrt 3}2$. But your formula gives $\sqrt{\frac12} = \frac{\sqrt2} 2$. – peterwhy Feb 20 '24 at 02:43
  • @peterwhy any ideas of case studies we could use to test the center formula? your case for the radius was fabulous. – Stan Shunpike Feb 20 '24 at 04:58