# Calculate 4th coordinate given dihedral angle and 3 coordinates

For a given dihedral angle and 3 coordinates, how do I calculate the coordinate of the 4th point that defines the dihedral angle? I suppose in some ways it's like the question here, but I'm calculating the coordinate of one point rather than the dihedral angle; How do I calculate a dihedral angle given Cartesian coordinates?

Never mind, this paper has the way to do it: http://onlinelibrary.wiley.com/doi/10.1002/prot.22488/full

As per my efforts, the fourth co-ordinate can not be found with the given data

let the points given be A, B, C & D of which let C be unknown

Let the given Dihedral angle is between the planes ABD & ADC, now in the figure above let BE & CF both be perpendiculars to AD extended.

(here the four given points should not be co-planer else dihedral angle will be 180 degrees)

Then the Dihedral angle is the angle between BE & blue line (which is parallel to FC)

Now as A, B, D & E are co-planer and A, B & D are known so E can be calculated..

And also we know the vector perpendicular to plane ABD so from dihedral angle, we can calculate the angle perpendicular to plane ADC and with points A & D, plane ADC can be uniquely found.

And we have

(unit vector in the direction of DA)*(tan {Dihedral angle }) = (cross product of FC & EB)/(dot product of FC & EB)

from here, vector FC can be calculated....

But either of F or C should be known to identify C...

Hence C can not be found with the given data alone...

Experts may comment.