Problem involving a tetrahedron

[NiGHTS]

This is a problem I am trying to solve for an engineering situation, but I am also very interested to learn the process involved to solve this problem.

My goal is to determine the diameter of a sphere using only the Y coordinates of three points of a tetrahedron. Below is an image showing an example.



The tetrahedron is formed by drawing three perpendicular lines from the center of a sphere to the edges of the sphere. The angle of this tetrahedron is unknown. All I am provided is the Y coordinates relative to the green vertical line of point A, B, and C. Point D is always (0,0).

Here is an example of coordinate data I am receiving: A = -23, B = -174, C = -466. This makes up three points directly related to the Y axis, so my interest is in knowing the size of the sphere contained in it.

My initial thought on how to tackle this problem is to use Euler angles in some way, unfortunately I'm lost on how to go about that for this kind of situation.

Thank you in advance for your help!

 

[NiGHTS]

I've figured out the solution on my own. I realised that this was a rotation calculation, much in the same way as it would be done on a 3D computer video game, except I had to calculate in a backwards direction.

I started with a 2D rotation function:

x' = x × cos(β) - y × sin(β)
y' = y × cos(β) + x × sin(β)

Then I tripled it, one for each pair of axes.

Then I defined my problem with numbers. My shape has 4 points of interest. (0,0,0); (0,0,n); (0,n,0); (n, 0, 0). My interest is in the value of 'n'.

Then I plugged in all my values into the array of equations. Doing so caused lots of multiplication of zeros which simplified the equations significantly.

To combine them into one equation was all about substitution. Once I finally simplified it all I came up with a surprising final equation.

n² = a² + b²

So I changed it up to include all three points of interest...

q² = x² + y²
n² = z² + q²

In the above equations, x, y, and z represents the Y coordinate of each of these origin axes points. q is a 'temp' variable if you will.

So that's how I can know what the radius of the circle around my shape is given only three Y axis coordinates.