# Source code for ase.geometry.cell

```# Copyright (C) 2010, Jesper Friis
# (see accompanying license files for details).

# XXX bravais objects need to hold tolerance eps, *or* temember variant
# from the beginning.
#
# Should they hold a 'cycle' argument or other data to reconstruct a particular
# cell?  (E.g. rotation, niggli transform)
#
# Implement total ordering of Bravais classes 1-14

import numpy as np
from numpy import pi, sin, cos, arccos, sqrt, dot
from numpy.linalg import norm

def unit_vector(x):
"""Return a unit vector in the same direction as x."""
y = np.array(x, dtype='float')
return y / norm(y)

def angle(x, y):
"""Return the angle between vectors a and b in degrees."""
return arccos(dot(x, y) / (norm(x) * norm(y))) * 180. / pi

"""Returns the cell parameters [a, b, c, alpha, beta, gamma].

Angles are in degrees unless radian=True is used.
"""
lengths = [np.linalg.norm(v) for v in cell]
angles = []
for i in range(3):
j = i - 1
k = i - 2
ll = lengths[j] * lengths[k]
if ll > 1e-16:
x = np.dot(cell[j], cell[k]) / ll
angle = 180.0 / pi * arccos(x)
else:
angle = 90.0
angles.append(angle)
angles = [angle * pi / 180 for angle in angles]
return np.array(lengths + angles)

[docs]def cellpar_to_cell(cellpar, ab_normal=(0, 0, 1), a_direction=None):
"""Return a 3x3 cell matrix from cellpar=[a,b,c,alpha,beta,gamma].

Angles must be in degrees.

The returned cell is orientated such that a and b
are normal to `ab_normal` and a is parallel to the projection of
`a_direction` in the a-b plane.

Default `a_direction` is (1,0,0), unless this is parallel to
`ab_normal`, in which case default `a_direction` is (0,0,1).

The returned cell has the vectors va, vb and vc along the rows. The
cell will be oriented such that va and vb are normal to `ab_normal`
and va will be along the projection of `a_direction` onto the a-b
plane.

Example:

>>> cell = cellpar_to_cell([1, 2, 4, 10, 20, 30], (0, 1, 1), (1, 2, 3))
>>> np.round(cell, 3)
array([[ 0.816, -0.408,  0.408],
[ 1.992, -0.13 ,  0.13 ],
[ 3.859, -0.745,  0.745]])

"""
if a_direction is None:
if np.linalg.norm(np.cross(ab_normal, (1, 0, 0))) < 1e-5:
a_direction = (0, 0, 1)
else:
a_direction = (1, 0, 0)

# Define rotated X,Y,Z-system, with Z along ab_normal and X along
# the projection of a_direction onto the normal plane of Z.
Z = unit_vector(ab_normal)
Y = np.cross(Z, X)

# Express va, vb and vc in the X,Y,Z-system
alpha, beta, gamma = 90., 90., 90.
if isinstance(cellpar, (int, float)):
a = b = c = cellpar
elif len(cellpar) == 1:
a = b = c = cellpar[0]
elif len(cellpar) == 3:
a, b, c = cellpar
else:
a, b, c, alpha, beta, gamma = cellpar

# Handle orthorhombic cells separately to avoid rounding errors
eps = 2 * np.spacing(90.0, dtype=np.float64)  # around 1.4e-14
# alpha
if abs(abs(alpha) - 90) < eps:
cos_alpha = 0.0
else:
cos_alpha = cos(alpha * pi / 180.0)
# beta
if abs(abs(beta) - 90) < eps:
cos_beta = 0.0
else:
cos_beta = cos(beta * pi / 180.0)
# gamma
if abs(gamma - 90) < eps:
cos_gamma = 0.0
sin_gamma = 1.0
elif abs(gamma + 90) < eps:
cos_gamma = 0.0
sin_gamma = -1.0
else:
cos_gamma = cos(gamma * pi / 180.0)
sin_gamma = sin(gamma * pi / 180.0)

# Build the cell vectors
va = a * np.array([1, 0, 0])
vb = b * np.array([cos_gamma, sin_gamma, 0])
cx = cos_beta
cy = (cos_alpha - cos_beta * cos_gamma) / sin_gamma
cz_sqr = 1. - cx * cx - cy * cy
assert cz_sqr >= 0
cz = sqrt(cz_sqr)
vc = c * np.array([cx, cy, cz])

# Convert to the Cartesian x,y,z-system
abc = np.vstack((va, vb, vc))
T = np.vstack((X, Y, Z))
cell = dot(abc, T)

return cell

def metric_from_cell(cell):
"""Calculates the metric matrix from cell, which is given in the
Cartesian system."""
cell = np.asarray(cell, dtype=float)
return np.dot(cell, cell.T)

[docs]def complete_cell(cell):
"""Calculate complete cell with missing lattice vectors.

Returns a new 3x3 ndarray.
"""

cell = np.array(cell, dtype=float)
missing = np.nonzero(~cell.any(axis=1))[0]

if len(missing) == 3:
cell.flat[::4] = 1.0
if len(missing) == 2:
# Must decide two vectors:
V, s, WT = np.linalg.svd(cell.T)
sf = [s[0], 1, 1]
cell = (V @ np.diag(sf) @ WT).T
if np.sign(np.linalg.det(cell)) < 0:
cell[missing[0]] = -cell[missing[0]]
elif len(missing) == 1:
i = missing[0]
cell[i] = np.cross(cell[i - 2], cell[i - 1])
cell[i] /= np.linalg.norm(cell[i])

return cell

[docs]def is_orthorhombic(cell):
"""Check that cell only has stuff in the diagonal."""
return not (np.flatnonzero(cell) % 4).any()

[docs]def orthorhombic(cell):
"""Return cell as three box dimensions or raise ValueError."""
if not is_orthorhombic(cell):
raise ValueError('Not orthorhombic')
return cell.diagonal().copy()

# We make the Cell object available for import from here for compatibility
from ase.cell import Cell  # noqa
```