def rotation_matrix(axis, theta, units="deg"):
'''
Obtain a left multiplication rotation matrix, given the axis and angle you
wish to rotate by. By default it assumes units of degrees. If theta is in
radians, set units to rad.
**Parameters**
axis: *list, float*
The axis in which to rotate around.
theta: *float*
The angle of rotation.
units: *str, optional*
The units of theta (deg or rad).
**Returns**
rotatation_matrix: *list, list, float*
The left multiplication rotation matrix.
**References**
* http://stackoverflow.com/questions/6802577/python-rotation-of-3d-vector/25709323#25709323
'''
cast.assert_vec(axis, length=3, numeric=True)
axis = np.array(axis)
assert cast.is_numeric(theta),\
"Error - Theta should be a numerical value (it is %s)." % str(theta)
theta = float(theta)
if "deg" in units.lower():
theta = np.radians(theta)
return scipy.linalg.expm(
np.cross(np.eye(3), axis / scipy.linalg.norm(axis) * theta))
def __init__(self,
name,
box_size=(10.0, 10.0, 10.0),
box_angles=(90.0, 90.0, 90.0),
periodic=False):
self.name = str(name)
self.periodic = periodic
self.atoms = []
self.bonds = []
self.angles = []
self.dihedrals = []
self.molecules = []
self.parameters = None
self.atom_labels = None
cast.assert_vec(box_size)
cast.assert_vec(box_angles)
self.box_size = np.array(list(map(float, box_size)))
self.box_angles = np.array(list(map(float, box_angles)))
a, b, c = self.box_size
alpha, beta, gamma = self.box_angles
EPS = 1E-3
if any([abs(ba - 90) > EPS for ba in box_angles]):
'''
If the angles are not 90, the system is triclinic. So we will set
the relative information here. For LAMMPS, trigonal vectors are
established by using xy, xz, and yz:
A = (xhi - xlo, 0.0, 0.0)
B = (xy, yhi - ylo, 0.0)
C = (xz, yz, zhi - zlo)
Formula for converting (a, b, c, alpha, beta, gamma) to
(lx, ly, lz, xy, xz, yz) taken from online lammps help.
'''
self.xlo, self.ylo, self.zlo = 0.0, 0.0, 0.0
self.xhi = a
self.xy = b * np.cos(np.deg2rad(gamma))
self.xz = c * np.cos(np.deg2rad(beta))
self.yhi = np.sqrt(b**2 - self.xy**2)
self.yz = (b * c * np.cos(np.deg2rad(alpha)) -
self.xy * self.xz) / self.yhi
self.zhi = np.sqrt(c**2 - self.xz**2 - self.yz**2)
else:
'''
Otherwise, we have a monoclinic box. We will set the
center to the euclidean origin.
'''
self.xlo = -a / 2.0
self.ylo = -b / 2.0
self.zlo = -c / 2.0
self.xhi = a / 2.0
self.yhi = b / 2.0
self.zhi = c / 2.0
self.xy = 0.0
self.xz = 0.0
self.yz = 0.0
def __sub__(self, other):
'''
If given some 3D array, offset all atomic coordinates.
**Parameters**
other: *array, float*
Some 3D array/tuple/list of sorts that indicates a
translation.
**Returns**
None
'''
cast.assert_vec(other, length=3, numeric=True)
return self.__add__(-np.array(other, dtype=float))
def scale(self, v):
'''
Apply a scalar to this molecule.
**Parameters**
v: *list, float*
A vector of 3 floats specifying the x, y,
and z scalars to be applied.
**Returns**
None
'''
cast.assert_vec(v, length=3, numeric=True)
for a in self.atoms:
a.scale(v)
def set_position(self, pos):
'''
Manually set the atomic positions by passing a tuple/list.
**Parameters**
pos: *list, float or tuple, float*
A vector of 3 floats specifying the new x, y, and z coordinate.
**Returns**
None
'''
cast.assert_vec(pos, length=3, numeric=True)
self.x = pos[0]
self.y = pos[1]
self.z = pos[2]
def translate(self, v):
'''
Apply a translation to this molecule.
**Parameters**
v: *list, float*
A vector of 3 floats specifying the x, y,
and z offsets to be applied.
**Returns**
None
'''
cast.assert_vec(v, length=3, numeric=True)
for a in self.atoms:
a.translate(v)
def __truediv__(self, other):
'''
If given some 3D array, scale all atomic coordinates.
**Parameters**
other: *array, float*
Some 3D array/tuple/list of sorts that indicates a
scalar operation.
**Returns**
None
'''
assert 0.0 not in other,\
"Error - Cannot divide by 0!"
cast.assert_vec(other, length=3, numeric=True)
return self.__mul__(np.array(1.0 / np.array(other, dtype=float)))
def __add__(self, other):
'''
If given some 3D array, offset all atomic coordinates.
**Parameters**
other: *array, float*
Some 3D array/tuple/list of sorts that indicates a
translation.
**Returns**
None
'''
cast.assert_vec(other, length=3, numeric=True)
new = copy.deepcopy(self)
new.translate(other)
return new
def translate(self, v):
'''
Translate the atom by a vector.
**Parameters**
v: *list, float*
A vector of 3 floats specifying the x, y,
and z offsets to be applied.
**Returns**
None
'''
cast.assert_vec(v, length=3, numeric=True)
self.x += float(v[0])
self.y += float(v[1])
self.z += float(v[2])
def __mul__(self, other):
'''
If given some 3D array, scale all atomic coordinates.
**Parameters**
other: *array, float*
Some 3D array/tuple/list of sorts that indicates a
scalar operation.
**Returns**
None
'''
cast.assert_vec(other, length=3, numeric=True)
new = copy.deepcopy(self)
for i, mol in enumerate(new.molecules):
new.molecules[i] = mol * other
return new
def scale(self, v):
'''
Scale the atom by a vector. This can be useful if we want to
change coordinate systems.
**Parameters**
v: *list, float*
A vector of 3 floats specifying the x, y,
and z scalars to be applied.
**Returns**
None
'''
cast.assert_vec(v, length=3, numeric=True)
self.x *= float(v[0])
self.y *= float(v[1])
self.z *= float(v[2])
def orthogonal_procrustes(A, ref_matrix, reflection=False):
'''
Using the orthogonal procrustes method, we find the unitary matrix R with
det(R) > 0 such that ||A*R - ref_matrix||^2 is minimized. This varies
from that within scipy by the addition of the reflection term, allowing
and disallowing inversion. NOTE - This means that the rotation matrix is
used for right side multiplication!
**Parameters**
A: *list,* :class:`squid.structures.atom.Atom`
A list of atoms for which R will minimize the frobenius
norm ||A*R - ref_matrix||^2.
ref_matrix: *list,* :class:`squid.structures.atom.Atom`
A list of atoms for which *A* is being rotated towards.
reflection: *bool, optional*
Whether inversion is allowed (True) or not (False).
**Returns**
R: *list, list, float*
Right multiplication rotation matrix to best overlay A onto the
reference matrix.
scale: *float*
Scalar between the matrices.
**Derivation**
Goal: minimize ||A\*R - ref||^2, switch to trace
trace((A\*R-ref).T\*(A\*R-ref)), now we distribute
trace(R'\*A'\*A\*R) + trace(ref.T\*ref) - trace((A\*R).T\*ref) -
trace(ref.T\*(A\*R)), trace doesn't care about order, so re-order
trace(R\*R.T\*A.T\*A) + trace(ref.T\*ref) - trace(R.T\*A.T\*ref) -
trace(ref.T\*A\*R), simplify
trace(A.T\*A) + trace(ref.T\*ref) - 2\*trace(ref.T\*A\*R)
Thus, to minimize we want to maximize trace(ref.T \* A \* R)
u\*w\*v.T = (ref.T\*A).T
ref.T \* A = w \* u.T \* v
trace(ref.T \* A \* R) = trace (w \* u.T \* v \* R)
differences minimized when trace(ref.T \* A \* R) is maximized, thus
when trace(u.T \* v \* R) is maximized
This occurs when u.T \* v \* R = I (as u, v and R are all unitary
matrices so max is 1)
R is a rotation matrix so R.T = R^-1
u.T \* v \* I = R^-1 = R.T
R = u \* v.T
Thus, R = u.dot(vt)
**References**
* https://github.com/scipy/scipy/blob/v0.16.0/scipy/linalg/
_procrustes.py#L14
* http://compgroups.net/comp.soft-sys.matlab/procrustes-analysis
-without-reflection/896635
'''
assert hasattr(A, "__len__") and hasattr(ref_matrix, "__len__"),\
"Error - A and ref_matrix must be lists of atomic coordinates!"
cast.assert_vec(A[0], length=3, numeric=True)
cast.assert_vec(ref_matrix[0], length=3, numeric=True)
A = np.asarray_chkfinite(A)
ref_matrix = np.asarray_chkfinite(ref_matrix)
if A.ndim != 2:
raise ValueError('expected ndim to be 2, but observed %s' % A.ndim)
if A.shape != ref_matrix.shape:
raise ValueError('the shapes of A and ref_matrix differ (%s vs %s)' %
(A.shape, ref_matrix.shape))
u, w, vt = svd(A.T.dot(ref_matrix))
R = u.dot(vt) # Get the rotation matrix, including reflections
if not reflection and scipy.linalg.det(R) < 0:
# To remove reflection, we change the sign of the rightmost column of
# u (or v) and the scalar associated
# with that column
u[:, -1] *= -1
w[-1] *= -1
R = u.dot(vt)
scale = w.sum() # Get the scaled difference
return R, scale
def __truediv__(self, other):
assert 0.0 not in other,\
"Error - Cannot divide by 0!"
cast.assert_vec(other, length=3, numeric=True)
return self * np.array(1.0 / np.array(other, dtype=float))
def __mul__(self, other):
cast.assert_vec(other, length=3, numeric=True)
local_atom = self.replicate()
local_atom.scale(other)
return local_atom
def __sub__(self, other):
cast.assert_vec(other, length=3, numeric=True)
return self + -np.array(other)
