Source code for pymatgen.util.coord

# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.


"""
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"
__copyright__ = "Copyright 2011, The Materials Project"
__version__ = "1.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = "shyuep@gmail.com"
__date__ = "Nov 27, 2011"

# array size threshold for looping instead of broadcasting
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 threading servers. 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)
[docs]def pbc_shortest_vectors(lattice, fcoords1, fcoords2, mask=None, 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. mask (boolean array): Mask of matches that are not allowed. 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 cuc.pbc_shortest_vectors(lattice, fcoords1, fcoords2, mask, 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
[docs]def is_coord_subset_pbc(subset, superset, atol=1e-8, mask=None): """ 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 mask (boolean array): Mask of matches that are not allowed. 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) if mask is not None: m = np.array(mask, dtype=np.int) 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) elif units == "radians": 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 if not found: 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()