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# coding: utf-8 

# Copyright (c) Pymatgen Development Team. 

# Distributed under the terms of the MIT License. 

 

from __future__ import division, unicode_literals 

 

import numpy as np 

import re 

from math import sin, cos, pi, sqrt 

import string 

 

from monty.json import MSONable 

 

""" 

This module provides classes that operate on points or vectors in 3D space. 

""" 

 

 

__author__ = "Shyue Ping Ong, Shyam Dwaraknath" 

__copyright__ = "Copyright 2011, The Materials Project" 

__version__ = "1.0" 

__maintainer__ = "Shyue Ping Ong" 

__email__ = "shyuep@gmail.com" 

__status__ = "Production" 

__date__ = "Sep 23, 2011" 

 

 

class SymmOp(MSONable): 

""" 

A symmetry operation in cartesian space. Consists of a rotation plus a 

translation. Implementation is as an affine transformation matrix of rank 4 

for efficiency. Read: http://en.wikipedia.org/wiki/Affine_transformation. 

 

.. attribute:: affine_matrix 

 

A 4x4 numpy.array representing the symmetry operation. 

""" 

 

def __init__(self, affine_transformation_matrix, tol=0.01): 

""" 

Initializes the SymmOp from a 4x4 affine transformation matrix. 

In general, this constructor should not be used unless you are 

transferring rotations. Use the static constructors instead to 

generate a SymmOp from proper rotations and translation. 

 

Args: 

affine_transformation_matrix (4x4 array): Representing an 

affine transformation. 

tol (float): Tolerance for determining if matrices are equal. 

""" 

affine_transformation_matrix = np.array(affine_transformation_matrix) 

if affine_transformation_matrix.shape != (4, 4): 

raise ValueError("Affine Matrix must be a 4x4 numpy array!") 

self.affine_matrix = affine_transformation_matrix 

self.tol = tol 

 

@staticmethod 

def from_rotation_and_translation( 

rotation_matrix=((1, 0, 0), (0, 1, 0), (0, 0, 1)), 

translation_vec=(0, 0, 0), tol=0.1): 

""" 

Creates a symmetry operation from a rotation matrix and a translation 

vector. 

 

Args: 

rotation_matrix (3x3 array): Rotation matrix. 

translation_vec (3x1 array): Translation vector. 

tol (float): Tolerance to determine if rotation matrix is valid. 

 

Returns: 

SymmOp object 

""" 

rotation_matrix = np.array(rotation_matrix) 

translation_vec = np.array(translation_vec) 

if rotation_matrix.shape != (3, 3): 

raise ValueError("Rotation Matrix must be a 3x3 numpy array.") 

if translation_vec.shape != (3,): 

raise ValueError("Translation vector must be a rank 1 numpy array " 

"with 3 elements.") 

affine_matrix = np.eye(4) 

affine_matrix[0:3][:, 0:3] = rotation_matrix 

affine_matrix[0:3][:, 3] = translation_vec 

return SymmOp(affine_matrix, tol) 

 

def __eq__(self, other): 

return np.allclose(self.affine_matrix, other.affine_matrix, 

atol=self.tol) 

 

def __hash__(self): 

return 7 

 

def __repr__(self): 

return self.__str__() 

 

def __str__(self): 

output = ["Rot:", str(self.affine_matrix[0:3][:, 0:3]), "tau", 

str(self.affine_matrix[0:3][:, 3])] 

return "\n".join(output) 

 

def operate(self, point): 

""" 

Apply the operation on a point. 

 

Args: 

point: Cartesian coordinate. 

 

Returns: 

Coordinates of point after operation. 

""" 

affine_point = np.array([point[0], point[1], point[2], 1]) 

return np.dot(self.affine_matrix, affine_point)[0:3] 

 

def operate_multi(self, points): 

""" 

Apply the operation on a list of points. 

 

Args: 

points: List of Cartesian coordinates 

 

Returns: 

Numpy array of coordinates after operation 

""" 

points = np.array(points) 

affine_points = np.concatenate( 

[points, np.ones(points.shape[:-1] + (1,))], axis=-1) 

return np.inner(affine_points, self.affine_matrix)[..., :-1] 

 

def apply_rotation_only(self, vector): 

""" 

Vectors should only be operated by the rotation matrix and not the 

translation vector. 

 

Args: 

vector (3x1 array): A vector. 

""" 

return np.dot(self.rotation_matrix, vector) 

 

def transform_tensor(self, tensor): 

""" 

Applies rotation portion to a tensor. Note that tensor has to be in 

full form, not the Voigt form. 

 

Args: 

tensor (numpy array): a rank n tensor 

 

Returns: 

Transformed tensor. 

""" 

dim = tensor.shape 

rank = len(dim) 

assert all([i == 3 for i in dim]) 

# Build einstein sum string 

lc = string.ascii_lowercase 

indices = lc[:rank], lc[rank:2 * rank] 

einsum_string = ','.join([a + i for a, i in zip(*indices)]) 

einsum_string += ',{}->{}'.format(*indices[::-1]) 

einsum_args = [self.rotation_matrix] * rank + [tensor] 

 

return np.einsum(einsum_string, *einsum_args) 

 

def are_symmetrically_related(self, point_a, point_b, tol=0.001): 

""" 

Checks if two points are symmetrically related. 

 

Args: 

point_a (3x1 array): First point. 

point_b (3x1 array): Second point. 

tol (float): Absolute tolerance for checking distance. 

 

Returns: 

True if self.operate(point_a) == point_b or vice versa. 

""" 

if np.allclose(self.operate(point_a), point_b, atol=tol): 

return True 

if np.allclose(self.operate(point_b), point_a, atol=tol): 

return True 

return False 

 

@property 

def rotation_matrix(self): 

""" 

A 3x3 numpy.array representing the rotation matrix. 

""" 

return self.affine_matrix[0:3][:, 0:3] 

 

@property 

def translation_vector(self): 

""" 

A rank 1 numpy.array of dim 3 representing the translation vector. 

""" 

return self.affine_matrix[0:3][:, 3] 

 

def __mul__(self, other): 

""" 

Returns a new SymmOp which is equivalent to apply the "other" SymmOp 

followed by this one. 

""" 

new_matrix = np.dot(self.affine_matrix, other.affine_matrix) 

return SymmOp(new_matrix) 

 

@property 

def inverse(self): 

""" 

Returns inverse of transformation. 

""" 

invr = np.linalg.inv(self.affine_matrix) 

return SymmOp(invr) 

 

@staticmethod 

def from_axis_angle_and_translation(axis, angle, angle_in_radians=False, 

translation_vec=(0, 0, 0)): 

""" 

Generates a SymmOp for a rotation about a given axis plus translation. 

 

Args: 

axis: The axis of rotation in cartesian space. For example, 

[1, 0, 0]indicates rotation about x-axis. 

angle (float): Angle of rotation. 

angle_in_radians (bool): Set to True if angles are given in 

radians. Or else, units of degrees are assumed. 

translation_vec: A translation vector. Defaults to zero. 

 

Returns: 

SymmOp for a rotation about given axis and translation. 

""" 

if isinstance(axis, (tuple, list)): 

axis = np.array(axis) 

 

if isinstance(translation_vec, (tuple, list)): 

vec = np.array(translation_vec) 

else: 

vec = translation_vec 

 

a = angle if angle_in_radians else angle * pi / 180 

cosa = cos(a) 

sina = sin(a) 

u = axis / np.linalg.norm(axis) 

r = np.zeros((3, 3)) 

r[0, 0] = cosa + u[0] ** 2 * (1 - cosa) 

r[0, 1] = u[0] * u[1] * (1 - cosa) - u[2] * sina 

r[0, 2] = u[0] * u[2] * (1 - cosa) + u[1] * sina 

r[1, 0] = u[0] * u[1] * (1 - cosa) + u[2] * sina 

r[1, 1] = cosa + u[1] ** 2 * (1 - cosa) 

r[1, 2] = u[1] * u[2] * (1 - cosa) - u[0] * sina 

r[2, 0] = u[0] * u[2] * (1 - cosa) - u[1] * sina 

r[2, 1] = u[1] * u[2] * (1 - cosa) + u[0] * sina 

r[2, 2] = cosa + u[2] ** 2 * (1 - cosa) 

 

return SymmOp.from_rotation_and_translation(r, vec) 

 

@staticmethod 

def from_origin_axis_angle(origin, axis, angle, angle_in_radians=False): 

""" 

Generates a SymmOp for a rotation about a given axis through an 

origin. 

 

Args: 

origin (3x1 array): The origin which the axis passes through. 

axis (3x1 array): The axis of rotation in cartesian space. For 

example, [1, 0, 0]indicates rotation about x-axis. 

angle (float): Angle of rotation. 

angle_in_radians (bool): Set to True if angles are given in 

radians. Or else, units of degrees are assumed. 

 

Returns: 

SymmOp. 

""" 

theta = angle * pi / 180 if not angle_in_radians else angle 

a = origin[0] 

b = origin[1] 

c = origin[2] 

u = axis[0] 

v = axis[1] 

w = axis[2] 

# Set some intermediate values. 

u2 = u * u 

v2 = v * v 

w2 = w * w 

cos_t = cos(theta) 

sin_t = sin(theta) 

l2 = u2 + v2 + w2 

l = sqrt(l2) 

 

# Build the matrix entries element by element. 

m11 = (u2 + (v2 + w2) * cos_t) / l2 

m12 = (u * v * (1 - cos_t) - w * l * sin_t) / l2 

m13 = (u * w * (1 - cos_t) + v * l * sin_t) / l2 

m14 = (a * (v2 + w2) - u * (b * v + c * w) + 

(u * (b * v + c * w) - a * (v2 + w2)) * cos_t + 

(b * w - c * v) * l * sin_t) / l2 

 

m21 = (u * v * (1 - cos_t) + w * l * sin_t) / l2 

m22 = (v2 + (u2 + w2) * cos_t) / l2 

m23 = (v * w * (1 - cos_t) - u * l * sin_t) / l2 

m24 = (b * (u2 + w2) - v * (a * u + c * w) + 

(v * (a * u + c * w) - b * (u2 + w2)) * cos_t + 

(c * u - a * w) * l * sin_t) / l2 

 

m31 = (u * w * (1 - cos_t) - v * l * sin_t) / l2 

m32 = (v * w * (1 - cos_t) + u * l * sin_t) / l2 

m33 = (w2 + (u2 + v2) * cos_t) / l2 

m34 = (c * (u2 + v2) - w * (a * u + b * v) + 

(w * (a * u + b * v) - c * (u2 + v2)) * cos_t + 

(a * v - b * u) * l * sin_t) / l2 

 

return SymmOp([[m11, m12, m13, m14], [m21, m22, m23, m24], 

[m31, m32, m33, m34], [0, 0, 0, 1]]) 

 

@staticmethod 

def reflection(normal, origin=(0, 0, 0)): 

""" 

Returns reflection symmetry operation. 

 

Args: 

normal (3x1 array): Vector of the normal to the plane of 

reflection. 

origin (3x1 array): A point in which the mirror plane passes 

through. 

 

Returns: 

SymmOp for the reflection about the plane 

""" 

# Normalize the normal vector first. 

n = np.array(normal, dtype=float) / np.linalg.norm(normal) 

 

u, v, w = n 

 

translation = np.eye(4) 

translation[0:3, 3] = -np.array(origin) 

 

xx = 1 - 2 * u ** 2 

yy = 1 - 2 * v ** 2 

zz = 1 - 2 * w ** 2 

xy = -2 * u * v 

xz = -2 * u * w 

yz = -2 * v * w 

mirror_mat = [[xx, xy, xz, 0], [xy, yy, yz, 0], [xz, yz, zz, 0], 

[0, 0, 0, 1]] 

 

if np.linalg.norm(origin) > 1e-6: 

mirror_mat = np.dot(np.linalg.inv(translation), 

np.dot(mirror_mat, translation)) 

return SymmOp(mirror_mat) 

 

@staticmethod 

def inversion(origin=(0, 0, 0)): 

""" 

Inversion symmetry operation about axis. 

 

Args: 

origin (3x1 array): Origin of the inversion operation. Defaults 

to [0, 0, 0]. 

 

Returns: 

SymmOp representing an inversion operation about the origin. 

""" 

mat = -np.eye(4) 

mat[3, 3] = 1 

mat[0:3, 3] = 2 * np.array(origin) 

return SymmOp(mat) 

 

@staticmethod 

def rotoreflection(axis, angle, origin=(0, 0, 0)): 

""" 

Returns a roto-reflection symmetry operation 

 

Args: 

axis (3x1 array): Axis of rotation / mirror normal 

angle (float): Angle in degrees 

origin (3x1 array): Point left invariant by roto-reflection. 

Defaults to (0, 0, 0). 

 

Return: 

Roto-reflection operation 

""" 

rot = SymmOp.from_origin_axis_angle(origin, axis, angle) 

refl = SymmOp.reflection(axis, origin) 

m = np.dot(rot.affine_matrix, refl.affine_matrix) 

return SymmOp(m) 

 

def as_dict(self): 

d = {"@module": self.__class__.__module__, 

"@class": self.__class__.__name__, 

"matrix": self.affine_matrix.tolist(), "tolerance": self.tol} 

return d 

 

def as_xyz_string(self): 

""" 

Returns a string of the form 'x, y, z', '-x, -y, z', 

'-y+1/2, x+1/2, z+1/2', etc. Only works for integer rotation matrices 

""" 

xyz = ['x', 'y', 'z'] 

strings = [] 

 

# test for invalid rotation matrix 

if not np.all(np.isclose(self.rotation_matrix, 

np.round(self.rotation_matrix))): 

raise ValueError('Rotation matrix must be integer') 

 

for r, t in zip(self.rotation_matrix, self.translation_vector): 

symbols = [] 

for val, axis in zip(r, xyz): 

val = int(round(val)) 

if val == 1: 

if symbols: 

symbols.append('+') 

symbols.append(axis) 

elif val == -1: 

symbols.append('-' + axis) 

elif val > 1: 

if symbols: 

symbols.append('+') 

symbols.append(str(val) + axis) 

elif val < -1: 

symbols.append(str(val) + axis) 

import fractions 

f = fractions.Fraction(float(t)).limit_denominator() 

if abs(f) > 1e-6: 

if f > 0: 

symbols.append('+') 

symbols.append(str(f)) 

strings.append("".join(symbols)) 

return ', '.join(strings) 

 

@staticmethod 

def from_xyz_string(xyz_string): 

""" 

Args: 

xyz_string: string of the form 'x, y, z', '-x, -y, z', 

'-2y+1/2, 3x+1/2, z-y+1/2', etc. 

Returns: 

SymmOp 

""" 

rot_matrix = np.zeros((3, 3)) 

trans = np.zeros(3) 

toks = xyz_string.strip().replace(" ", "").lower().split(",") 

re_rot = re.compile("([+-]?)([\d\.]*)/?([\d\.]*)([x-z])") 

re_trans = re.compile("([+-]?)([\d\.]+)/?([\d\.]*)(?![x-z])") 

for i, tok in enumerate(toks): 

# build the rotation matrix 

for m in re_rot.finditer(tok): 

factor = -1 if m.group(1) == "-" else 1 

if m.group(2) != "": 

factor *= float(m.group(2)) / float(m.group(3)) \ 

if m.group(3) != "" else float(m.group(2)) 

j = ord(m.group(4)) - 120 

rot_matrix[i, j] = factor 

# build the translation vector 

for m in re_trans.finditer(tok): 

factor = -1 if m.group(1) == "-" else 1 

num = float(m.group(2)) / float(m.group(3)) \ 

if m.group(3) != "" else float(m.group(2)) 

trans[i] = num * factor 

return SymmOp.from_rotation_and_translation(rot_matrix, trans) 

 

@classmethod 

def from_dict(cls, d): 

return cls(d["matrix"], d["tolerance"])