Another math (geometry) question

[Osede]

This probably can consider my course project problems. But I have solved it in numerical way, just curious if there are any better analytical way (or closed form solution) for this.

The problem is as follow, we have a tetrahedron shape (see attachment), where none of the side of triangles are equal length. And the information we know are, r1,r2,r3 in unit vector (aka the direction, thus we know the theta1,theta2 and theta3), and the length of the base triangle, a,b,c. And we need to find the length of the r1,r2,r3 (which write as R1,R2,R3 in the numerical solution attached).

The nonlinear solution is shown in the attachment and it can be solved using any iteration method, e.g. newton raphson. But, I just wonder that if there is any closed form solution for this kind of tetrahedron problems. I found a paper that proposed the closed form solution for this, but when I try to revisit the method, I'm unable to obtained the solution (which sort of expected).

So, anyone in BG knows the other keyword that I could look at to find the closed form solution?

[Bugpop]

My best suggestion is to try using another coordinate system. Spherical looks promising.

[Max™]

My best suggestion is to try using another coordinate system. Spherical looks promising.

This, shouldn't be too hard to work it out from a spherical coordinate system, it's just a volume from a sphere with the bottom triangle ends touching the interior. Can work out the various sin/csc/tan etc from that and it should solve itself from there as I recall.

[Osede]

The paper I obtained was using spherical way to solve it. But just like to point out that only one point of the triangle will touch the sphere surface but they all unequal length.


But seems like spherical still one of the best way to do it...

[Max™]

Perhaps have the points which are not touching the surface be external so the triangle is cut by the sphere?