# Test if 2 transformations produce equivalent relations to a reference

## Main Question or Discussion Point

Hello --

I have some reference object R (e.g. a protein), and I've got two transformations t1 and t2 (e.g. a transformation = quaternion + translation). In my case, t1 and t2 were obtained from symmetry operations.

So after applying t1 to R I get object T1, and after applying t2 to R I get object T2. How do I determine whether system S1=R+T1 is equivalent to system S2=R+T2? That is, after eliminating the 6 laboratory degrees of freedom, whether S1 =?= S2?

The only way I can think of is to actually make S1 and S2 (or a subset of their points), minimize the root-mean-square-deviation of coordinates in S1 vs S2, and see if the rmsd == 0. However, my intuition says there should be something I can test in t1 vs t2 to determine this.

For example, I know that S1 and S2 are not equivalent if the magnitude of the two translations are different. Here's some sample data:

0[ -0.02845, -0.11515, -0.48573, 0.86603][ -30.36901, 16.88513, -10.19267][ -44.14951, 9.93415, -7.73766] dist = 15.6283439198
1[ -0.04928, -0.19945, -0.84130, 0.50000][ -30.36901, 16.88513, -10.19267][ -57.37728, 18.17323, -8.91610] dist = 27.0690857072
2[ -0.05690, -0.23030, -0.97145, 0.00000][ -30.36901, 16.88513, -10.19267][ -56.82456, 33.36329, -12.54953] dist = 31.2566878395
3[ -0.04928, -0.19945, -0.84130, -0.50000][ -30.36901, 16.88513, -10.19267][ -43.04406, 40.31426, -15.00454] dist = 27.0690857072
4[ -0.02845, -0.11515, -0.48573, -0.86603][ -30.36901, 16.88513, -10.19267][ -29.81629, 32.07518, -13.82610] dist = 15.6283439198

which represents a system describable as a C6 cyclic rotation, one row for each transformed unit. The first brackets have the quaternion (x,y,z,w), the second brackets are the pre-rotation translation, and the third brackets are the post-rotation translation. Lastly, I went ahead and computed the total translation distance = mag(post-pre). So looking at the distances, you can immediately guess that 0 and 4 are the units adjacent to the reference (i.e. ortho), 1 and 3 are meta, and 2 is para. The relationship between the reference and both ortho units have the equivalence I'm looking for, while the reference and the meta units also have that equivalence. The reason I want to know this is because if S1 and S2 are the same, I only need to calculate the energy for one of them and multiply it by 2.

For Cn, I already know which transformations are equivalent, but since I'm looking into implementing more complicated symmetry groups, I'm interested in simply detecting equivalence.

To see what I mean by ortho, meta, para:
http://en.wikipedia.org/wiki/Arene_substitution_patterns