amino acid dihedral angles from distances between 4 points (atoms)

Question

I wish to compute the angles between 4 atoms from the distances between them. I can use the law of cosines to work out the two simple triangles, but I am struggling to find a (hopefully elegant) solution for the dihedral.

The ASCII art below describes my preferred system: triangle ABC is on the XZ plane (Y=0), with B on the origin and C on the +Z axis. Triangle BCD rotates D around the Z axis, so x and y coordinates for D are the unknown values (but my current goal is the dihedral angle, not the coordinates):

        |
        |  D
   \    | /
    \   |/
     \  C
      \ I
       \I
--------B---------- x
       /|\
      / | \
     A  |  \
        |   \
        z    y

My goal now is to get the dihedral angle rotation for ABCD from the distances AB, BC, CD, AC, BD and AD.

I expect there will be a chirality issue (2 points on the circle traced by D about the Z axis with the same AD distance), but I believe I can solve this with other constraints.

Helpful direction, advice or implementation detail you may have will be very much appreciated.

Edit: I have implemented the algorithm referenced in the comments in Biopython version 1.80/1.81, specifically the distance_to_internal_coordinates() routine. I was unable to resolve the chirality issue and require this data as an input.

dihedral angle are angle between two planes. Are you asking for the angle between the planes holding $ABC$ and $BCD$? — achille hui, Sep 27 '21 at 08:17
Related (duplicate?): "The Dihedral Angles of a Tetrahedron in terms of its edge lengths". See, in particular, my answer. — Blue, Sep 27 '21 at 11:40
@achillehui - yes, the angle between the planes holding ABC and BCD along the edge BC — robm, Sep 27 '21 at 13:31
@Blue YES! I am pleased to report that your solution works for me (with the +/- chirality issue expected). Thank You So Much! I'm also pleased to note that I would not have come up with anything lke that solution, nor recognised the relationship to the earlier question had I not asked. — robm, Sep 27 '21 at 18:29

amino acid dihedral angles from distances between 4 points (atoms)

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