Finding azimuth and zenith angles for tetrahedron

Question

I have an irregular tetrahedron as shown in figure below, and I know the length of all of its edges. I want to find the azimuth and zenith angles $\phi$, and $\theta$ for vector $C$.

I managed to find the zenith angle $\theta$ by finding the volume using Cayley-Menger Determinant and the equation volume = $(A\times B).C$; however I can't find the azimuth angle $\theta$.

Any help is appreciated to find azimuth angle $\phi$.

Best regards,

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score 1 · Answer 1 · answered Sep 17 '22 at 11:16

1

Call $x$ the axis containing the edge $OA$, $z$ the axis wrt the polar angle $\theta$ is measured, $y$ the remaining axis and $\bf c$ the vector $OC$. Knowing the edges, by the cosines rule you can determine the angle $AOC$.
So you have $c_z, c_x$ and can subsequently determine $c_y$ from the $\left\| {\bf c} \right\|$.

answered Sep 17 '22 at 11:16

G Cab

35,272

Thanks, this is brilliant. however I can't determine $c_y$ has a negative value!! Do have an idea how to solve that? – Mahmoud Alnaanah Sep 17 '22 at 19:16
which are the three facet angles in O ? – G Cab Sep 17 '22 at 20:15

Finding azimuth and zenith angles for tetrahedron

1 Answers1