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There is a trivial case of this which I can intuitively understand.

Assuming the tetrahedron is inscribed in a unit sphere and the given points are $$v_1 = (1, 0°, x°)\\v_2 \approx (1, 109.5°, x°)$$ where $x$ is any azimuthal angle, then the remaining two are easily calculated as $$v_3 \approx (1, ~109.5°, x° + 120°)\\v_4 \approx (1, 109.5°, x° - 120°)$$ modulo the 3rd component by 360° if necessary.

I'm trying to generalize my implicit understanding of the above to work with any two given points, but am having some difficulty coming up with a generalized equation. Probably something involving Euler's rotation theorem, but I'm not quite sure how to implement that.

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    $\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$ Commented Apr 12 at 6:57
  • $\begingroup$ Thanks! Updated the post to make it more legible, I think. $\endgroup$ Commented Apr 12 at 7:16
  • $\begingroup$ It will be easier if you use vector. if $u$,$v$ are the two given vertices, the two other vertices can be obtained by rotating them wrt their mid-point (in the plane perpendicular to mid-point) for $90^\circ$ and then take their negative. This mean they have the form $-\frac{u+v}{2} \pm C u\times v$ for some constant $C$. Since they lie on the unit sphere, it is not hard to show $C = \frac{\sqrt{3}}{2}$. $\endgroup$ Commented Apr 12 at 8:04

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