C2v symmetry operations has the mirror plane around the wrong axis

[argerlt]

I need a sanity check on this please. Looking at ITOC and "Structure of Materials" (DeGraef 2012), I believe the symmetry group C2v in orix is incorrect:

>>> C2v
Symmetry (4,) mm2
[[ 1.  0.  0.  0.]
 [ 0.  1.  0.  0.]
 [ 0.  0.  0.  1.]
 [ 0.  0. -1.  0.]]

This is generated using Symmetry.from_generators(C2x,Csz) (2-fold around x, mirror around z, which creates mirror around y)

I think it should be generated instead with Symmetry.from_generators(C2z,Csx), so the 2-fold is around the z axis, as is the ITOC standard.

Symmetry (4,) mm2
[[1. 0. 0. 0.]
 [0. 0. 0. 1.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]]

(note in both, the last two operators are improper, and thus mirrors.)

ITOC page below for reference. Note how in the following snippet, the first set of vectors do not match ITOC, but the second do

from orix.quaternion.symmetry import C2z, C2x, Csx, Csz, Symmetry
from orix.vector import Vector3d

v = Vector3d([1,2,3])
Symmetry.from_generators(C2z,Csx)*v
Out[90]: 
Vector3d (4,)
[[ 1.  2.  3.]
 [-1. -2.  3.]
 [-1.  2.  3.]
 [ 1. -2.  3.]]

C2v*v
Out[91]: 
Vector3d (4,)
[[ 1.  2.  3.]
 [ 1. -2. -3.]
 [ 1.  2. -3.]
 [ 1. -2.  3.]]

I can take care of this if it is a mistake, I just want someone to check me before I accidentially waste time on a misunderstanding
Thank you.

[CSSFrancis]

@maclariz or @hakonanes any chance you can comment on this? @argerlt have you checked the other symmetries as well? Is this something that we could hardcode a sanity check using ITOC?

[argerlt]

have you checked the other symmetries as well?

Yes, I did go though and check all the point groups, line by line, with a physical copy of ITOC volA next to me. C2v is the only PG I found whose elements didn't match up with ITOC.

As a final exhaustive check, as part of #563, I wrote code to proceedurally generate all 32 stereographic plots, and they do, in fact, match ITOC, which seems about as close to verification as we can get.

Is this something that we could hardcode a sanity check using ITOC?

The only additional check I can think of would be to check that the highest order inversion/rotoinversion is always around z. Other than that, we do have hard-coded checks, it's just this one in particular was check with a flipped axes assumption.

[CSSFrancis]

@argerlt That seems good enough for me :) Thanks for doing that! It's tedious work, I think I owe you a beer after that :)

[hakonanes]

Good catch, @argerlt.

The vector (1, 2, 3) rotated with the four symmetry operations correspond to 2mm, so 211, i.e. a two-fold rotation around x, as you say:

(x, y, -z)
(x, -y, z)
(x, -y, -z)
(x, y, z)

[hakonanes]

How the symmetry operations of a point group rotate a vector with unique components (x, y, z) is actually a good test. We should consider add some of these.

[hakonanes]

This should be fixed in #601.