Given three vectors $u,v,w \in S^2$ and the triangle $[u,v,w]$ I want to find its circumscribed circle. However, I don't know how to approach this problem. Would some one please explain?

In my understanding, one needs to somehow find the altitudes of the spherical triangle and then the point of its intersection (circumcentre)? Then what about the radius?


The three points lie on a plane, which cuts the sphere in a circle. That is the circumscribed circle.

To calculate its radius, calculate the distance $d$ of the plane from the center of the sphere. If the sphere has radius $1$, then the radius of the circumcircle is $\arccos{(d)}$.

  • $\begingroup$ Can you please clarify why the radius of the circumcircle is $\arccos(d)$? I drew a picture of a sphere and a cutting plane, marked the radius of the section circle as $a$, the distance to the plane from the center as $d$ and the hypotenuse of the resulting right triangle as $h$, and the angle between $d$ and $h$ as $\theta$. Then I tried to express $a$ and $d$ in terms of $\cos\theta$, but it is only possible in terms of $\cot\theta$ or $\tan\theta$, as it seems. And that doesn't give much insight into obtaining an expression not involving $\theta$. $\endgroup$ – sequence Jun 24 '17 at 10:59
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    $\begingroup$ ibb.co/i1Sm8Q $\endgroup$ – MaudPieTheRocktorate Jun 24 '17 at 15:12

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