# Source code for pymatgen.util.coord

# coding: utf-8
# Copyright (c) Pymatgen Development Team.

"""
Utilities for manipulating coordinates or list of coordinates, under periodic
boundary conditions or otherwise. Many of these are heavily vectorized in
numpy for performance.
"""

import itertools
import numpy as np
import math
from . import coord_cython as cuc

__author__ = "Shyue Ping Ong"
__version__ = "1.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = "shyuep@gmail.com"
__date__ = "Nov 27, 2011"

LOOP_THRESHOLD = 1e6

[docs]def find_in_coord_list(coord_list, coord, atol=1e-8):
"""
Find the indices of matches of a particular coord in a coord_list.

Args:
coord_list: List of coords to test
coord: Specific coordinates
atol: Absolute tolerance. Defaults to 1e-8. Accepts both scalar and
array.

Returns:
Indices of matches, e.g., [0, 1, 2, 3]. Empty list if not found.
"""
if len(coord_list) == 0:
return []
diff = np.array(coord_list) - np.array(coord)[None, :]
return np.where(np.all(np.abs(diff) < atol, axis=1))[0]

[docs]def in_coord_list(coord_list, coord, atol=1e-8):
"""
Tests if a particular coord is within a coord_list.

Args:
coord_list: List of coords to test
coord: Specific coordinates
atol: Absolute tolerance. Defaults to 1e-8. Accepts both scalar and
array.

Returns:
True if coord is in the coord list.
"""
return len(find_in_coord_list(coord_list, coord, atol=atol)) > 0

[docs]def is_coord_subset(subset, superset, atol=1e-8):
"""
Tests if all coords in subset are contained in superset.
Doesn't use periodic boundary conditions

Args:
subset, superset: List of coords

Returns:
True if all of subset is in superset.
"""
c1 = np.array(subset)
c2 = np.array(superset)
is_close = np.all(np.abs(c1[:, None, :] - c2[None, :, :]) < atol, axis=-1)
any_close = np.any(is_close, axis=-1)
return np.all(any_close)

[docs]def coord_list_mapping(subset, superset, atol=1e-8):
"""
Gives the index mapping from a subset to a superset.
Subset and superset cannot contain duplicate rows

Args:
subset, superset: List of coords

Returns:
list of indices such that superset[indices] = subset
"""
c1 = np.array(subset)
c2 = np.array(superset)
inds = np.where(np.all(np.isclose(c1[:, None, :], c2[None, :, :], atol=atol),
axis=2))[1]
result = c2[inds]
if not np.allclose(c1, result, atol=atol):
if not is_coord_subset(subset, superset):
raise ValueError("subset is not a subset of superset")
if not result.shape == c1.shape:
raise ValueError("Something wrong with the inputs, likely duplicates "
"in superset")
return inds

[docs]def coord_list_mapping_pbc(subset, superset, atol=1e-8):
"""
Gives the index mapping from a subset to a superset.
Superset cannot contain duplicate matching rows

Args:
subset, superset: List of frac_coords

Returns:
list of indices such that superset[indices] = subset
"""
atol = np.array([1., 1., 1.]) * atol
return cuc.coord_list_mapping_pbc(subset, superset, atol)

[docs]def get_linear_interpolated_value(x_values, y_values, x):
"""
Returns an interpolated value by linear interpolation between two values.
This method is written to avoid dependency on scipy, which causes issues on

Args:
x_values: Sequence of x values.
y_values: Corresponding sequence of y values
x: Get value at particular x

Returns:
Value at x.
"""
a = np.array(sorted(zip(x_values, y_values), key=lambda d: d[0]))

ind = np.where(a[:, 0] >= x)[0]

if len(ind) == 0 or ind[0] == 0:
raise ValueError("x is out of range of provided x_values")

i = ind[0]
x1, x2 = a[i - 1][0], a[i][0]
y1, y2 = a[i - 1][1], a[i][1]

return y1 + (y2 - y1) / (x2 - x1) * (x - x1)

[docs]def all_distances(coords1, coords2):
"""
Returns the distances between two lists of coordinates

Args:
coords1: First set of cartesian coordinates.
coords2: Second set of cartesian coordinates.

Returns:
2d array of cartesian distances. E.g the distance between
coords1[i] and coords2[j] is distances[i,j]
"""
c1 = np.array(coords1)
c2 = np.array(coords2)
z = (c1[:, None, :] - c2[None, :, :]) ** 2
return np.sum(z, axis=-1) ** 0.5

[docs]def pbc_diff(fcoords1, fcoords2):
"""
Returns the 'fractional distance' between two coordinates taking into
account periodic boundary conditions.

Args:
fcoords1: First set of fractional coordinates. e.g., [0.5, 0.6,
0.7] or [[1.1, 1.2, 4.3], [0.5, 0.6, 0.7]]. It can be a single
coord or any array of coords.
fcoords2: Second set of fractional coordinates.

Returns:
Fractional distance. Each coordinate must have the property that
abs(a) <= 0.5. Examples:
pbc_diff([0.1, 0.1, 0.1], [0.3, 0.5, 0.9]) = [-0.2, -0.4, 0.2]
pbc_diff([0.9, 0.1, 1.01], [0.3, 0.5, 0.9]) = [-0.4, -0.4, 0.11]
"""
fdist = np.subtract(fcoords1, fcoords2)
return fdist - np.round(fdist)

return_d2=False):
"""
Returns the shortest vectors between two lists of coordinates taking into
account periodic boundary conditions and the lattice.

Args:
lattice: lattice to use
fcoords1: First set of fractional coordinates. e.g., [0.5, 0.6, 0.7]
or [[1.1, 1.2, 4.3], [0.5, 0.6, 0.7]]. It can be a single
coord or any array of coords.
fcoords2: Second set of fractional coordinates.
i.e. if mask[1,2] == True, then subset[1] cannot be matched
to superset[2]
return_d2 (boolean): whether to also return the squared distances

Returns:
array of displacement vectors from fcoords1 to fcoords2
first index is fcoords1 index, second is fcoords2 index
"""
return_d2)

[docs]def find_in_coord_list_pbc(fcoord_list, fcoord, atol=1e-8):
"""
Get the indices of all points in a fractional coord list that are
equal to a fractional coord (with a tolerance), taking into account
periodic boundary conditions.

Args:
fcoord_list: List of fractional coords
fcoord: A specific fractional coord to test.
atol: Absolute tolerance. Defaults to 1e-8.

Returns:
Indices of matches, e.g., [0, 1, 2, 3]. Empty list if not found.
"""
if len(fcoord_list) == 0:
return []
fcoords = np.tile(fcoord, (len(fcoord_list), 1))
fdist = fcoord_list - fcoords
fdist -= np.round(fdist)
return np.where(np.all(np.abs(fdist) < atol, axis=1))[0]

[docs]def in_coord_list_pbc(fcoord_list, fcoord, atol=1e-8):
"""
Tests if a particular fractional coord is within a fractional coord_list.

Args:
fcoord_list: List of fractional coords to test
fcoord: A specific fractional coord to test.
atol: Absolute tolerance. Defaults to 1e-8.

Returns:
True if coord is in the coord list.
"""
return len(find_in_coord_list_pbc(fcoord_list, fcoord, atol=atol)) > 0

"""
Tests if all fractional coords in subset are contained in superset.

Args:
subset, superset: List of fractional coords
atol (float or size 3 array): Tolerance for matching
i.e. if mask[1,2] == True, then subset[1] cannot be matched
to superset[2]

Returns:
True if all of subset is in superset.
"""
c1 = np.array(subset, dtype=np.float64)
c2 = np.array(superset, dtype=np.float64)
else:
m = np.zeros((len(subset), len(superset)), dtype=np.int)
atol = np.zeros(3, dtype=np.float64) + atol
return cuc.is_coord_subset_pbc(c1, c2, atol, m)

[docs]def lattice_points_in_supercell(supercell_matrix):
"""
Returns the list of points on the original lattice contained in the
supercell in fractional coordinates (with the supercell basis).
e.g. [[2,0,0],[0,1,0],[0,0,1]] returns [[0,0,0],[0.5,0,0]]

Args:
supercell_matrix: 3x3 matrix describing the supercell

Returns:
numpy array of the fractional coordinates
"""
diagonals = np.array(
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1],
[1, 1, 0], [1, 1, 1]])
d_points = np.dot(diagonals, supercell_matrix)

mins = np.min(d_points, axis=0)
maxes = np.max(d_points, axis=0) + 1

ar = np.arange(mins[0], maxes[0])[:, None] * np.array([1, 0, 0])[None, :]
br = np.arange(mins[1], maxes[1])[:, None] * np.array([0, 1, 0])[None, :]
cr = np.arange(mins[2], maxes[2])[:, None] * np.array([0, 0, 1])[None, :]

all_points = ar[:, None, None] + br[None, :, None] + cr[None, None, :]
all_points = all_points.reshape((-1, 3))

frac_points = np.dot(all_points, np.linalg.inv(supercell_matrix))

tvects = frac_points[np.all(frac_points < 1 - 1e-10, axis=1)
& np.all(frac_points >= -1e-10, axis=1)]
assert len(tvects) == round(abs(np.linalg.det(supercell_matrix)))
return tvects

[docs]def barycentric_coords(coords, simplex):
"""
Converts a list of coordinates to barycentric coordinates, given a
simplex with d+1 points. Only works for d >= 2.

Args:
coords: list of n coords to transform, shape should be (n,d)
simplex: list of coordinates that form the simplex, shape should be
(d+1, d)

Returns:
a LIST of barycentric coordinates (even if the original input was 1d)
"""
coords = np.atleast_2d(coords)

t = np.transpose(simplex[:-1, :]) - np.transpose(simplex[-1, :])[:, None]
all_but_one = np.transpose(
np.linalg.solve(t, np.transpose(coords - simplex[-1])))
last_coord = 1 - np.sum(all_but_one, axis=-1)[:, None]
return np.append(all_but_one, last_coord, axis=-1)

[docs]def get_angle(v1, v2, units="degrees"):
"""
Calculates the angle between two vectors.

Args:
v1: Vector 1
v2: Vector 2
units: "degrees" or "radians". Defaults to "degrees".

Returns:
Angle between them in degrees.
"""
d = np.dot(v1, v2) / np.linalg.norm(v1) / np.linalg.norm(v2)
d = min(d, 1)
d = max(d, -1)
angle = math.acos(d)
if units == "degrees":
return math.degrees(angle)
return angle
else:
raise ValueError("Invalid units {}".format(units))

[docs]class Simplex:
"""
A generalized simplex object. See http://en.wikipedia.org/wiki/Simplex.

.. attribute: space_dim

Dimension of the space. Usually, this is 1 more than the simplex_dim.

.. attribute: simplex_dim

Dimension of the simplex coordinate space.
"""

def __init__(self, coords):
"""
Initializes a Simplex from vertex coordinates.

Args:
coords ([[float]]): Coords of the vertices of the simplex. E.g.,
[[1, 2, 3], [2, 4, 5], [6, 7, 8], [8, 9, 10].
"""
self._coords = np.array(coords)
self.space_dim, self.simplex_dim = self._coords.shape
self.origin = self._coords[-1]
if self.space_dim == self.simplex_dim + 1:
# precompute augmented matrix for calculating bary_coords
self._aug = np.concatenate([coords, np.ones((self.space_dim, 1))],
axis=-1)
self._aug_inv = np.linalg.inv(self._aug)

@property
def volume(self):
"""
Volume of the simplex.
"""
return abs(np.linalg.det(self._aug)) / math.factorial(self.simplex_dim)

[docs]    def bary_coords(self, point):
"""
Args:
point (): Point coordinates.

Returns:
Barycentric coordinations.
"""
try:
return np.dot(np.concatenate([point, [1]]), self._aug_inv)
except AttributeError:
raise ValueError('Simplex is not full-dimensional')

[docs]    def point_from_bary_coords(self, bary_coords):
"""
Args:
bary_coords (): Barycentric coordinates

Returns:
Point coordinates
"""
try:
return np.dot(bary_coords, self._aug[:, :-1])
except AttributeError:
raise ValueError('Simplex is not full-dimensional')

[docs]    def in_simplex(self, point, tolerance=1e-8):
"""
Checks if a point is in the simplex using the standard barycentric
coordinate system algorithm.

Taking an arbitrary vertex as an origin, we compute the basis for the
simplex from this origin by subtracting all other vertices from the
origin. We then project the point into this coordinate system and
determine the linear decomposition coefficients in this coordinate
system.  If the coeffs satisfy that all coeffs >= 0, the composition
is in the facet.

Args:
point ([float]): Point to test
tolerance (float): Tolerance to test if point is in simplex.
"""
return (self.bary_coords(point) >= -tolerance).all()

[docs]    def line_intersection(self, point1, point2, tolerance=1e-8):
"""
Computes the intersection points of a line with a simplex
Args:
point1, point2 ([float]): Points that determine the line
Returns:
points where the line intersects the simplex (0, 1, or 2)
"""
b1 = self.bary_coords(point1)
b2 = self.bary_coords(point2)
l = b1 - b2
# don't use barycentric dimension where line is parallel to face
valid = np.abs(l) > 1e-10
# array of all the barycentric coordinates on the line where
# one of the values is 0
possible = b1 - (b1[valid] / l[valid])[:, None] * l
barys = []
for p in possible:
# it's only an intersection if its in the simplex
if (p >= -tolerance).all():
found = False
# don't return duplicate points
for b in barys:
if np.all(np.abs(b - p) < tolerance):
found = True
break
barys.append(p)
assert len(barys) < 3
return [self.point_from_bary_coords(b) for b in barys]

def __eq__(self, other):
for p in itertools.permutations(self._coords):
if np.allclose(p, other.coords):
return True
return False

def __hash__(self):
return len(self._coords)

def __repr__(self):
output = ["{}-simplex in {}D space".format(self.simplex_dim,
self.space_dim),
"Vertices:"]
for coord in self._coords:
output.append("\t({})".format(", ".join(map(str, coord))))
return "\n".join(output)

def __str__(self):
return self.__repr__()

@property
def coords(self):
"""
Returns a copy of the vertex coordinates in the simplex.
"""
return self._coords.copy()