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# MDAnalysis --- https://www.mdanalysis.org
# Copyright (c) 2006-2017 The MDAnalysis Development Team and contributors
# (see the file AUTHORS for the full list of names)
#
# Released under the GNU Public Licence, v2 or any higher version
#
# Please cite your use of MDAnalysis in published work:
#
# R. J. Gowers, M. Linke, J. Barnoud, T. J. E. Reddy, M. N. Melo, S. L. Seyler,
# D. L. Dotson, J. Domanski, S. Buchoux, I. M. Kenney, and O. Beckstein.
# MDAnalysis: A Python package for the rapid analysis of molecular dynamics
# simulations. In S. Benthall and S. Rostrup editors, Proceedings of the 15th
# Python in Science Conference, pages 102-109, Austin, TX, 2016. SciPy.
# doi: 10.25080/majora-629e541a-00e
#
# N. Michaud-Agrawal, E. J. Denning, T. B. Woolf, and O. Beckstein.
# MDAnalysis: A Toolkit for the Analysis of Molecular Dynamics Simulations.
# J. Comput. Chem. 32 (2011), 2319--2327, doi:10.1002/jcc.21787
#
"""
Mathematical helper functions --- :mod:`MDAnalysis.lib.mdamath`
===============================================================
Helper functions for common mathematical operations
.. autofunction:: normal
.. autofunction:: norm
.. autofunction:: angle
.. autofunction:: dihedral
.. autofunction:: stp
.. autofunction:: sarrus_det
.. autofunction:: triclinic_box
.. autofunction:: triclinic_vectors
.. autofunction:: box_volume
.. autofunction:: make_whole
.. autofunction:: find_fragments
.. versionadded:: 0.11.0
.. versionchanged: 1.0.0
Unused function :func:`_angle()` has now been removed.
"""
import numpy as np
from ..exceptions import NoDataError
from . import util
from ._cutil import (make_whole, find_fragments, _sarrus_det_single,
_sarrus_det_multiple)
# geometric functions
[docs]def norm(v):
r"""Calculate the norm of a vector v.
.. math:: v = \sqrt{\mathbf{v}\cdot\mathbf{v}}
This version is faster then numpy.linalg.norm because it only works for a
single vector and therefore can skip a lot of the additional fuss
linalg.norm does.
Parameters
----------
v : array_like
1D array of shape (N) for a vector of length N
Returns
-------
float
norm of the vector
"""
return np.sqrt(np.dot(v, v))
[docs]def normal(vec1, vec2):
r"""Returns the unit vector normal to two vectors.
.. math::
\hat{\mathbf{n}} = \frac{\mathbf{v}_1 \times \mathbf{v}_2}{|\mathbf{v}_1 \times \mathbf{v}_2|}
If the two vectors are collinear, the vector :math:`\mathbf{0}` is returned.
.. versionchanged:: 0.11.0
Moved into lib.mdamath
"""
normal = np.cross(vec1, vec2)
n = norm(normal)
if n == 0.0:
return normal # returns [0,0,0] instead of [nan,nan,nan]
# ... could also use numpy.nan_to_num(normal/norm(normal))
return normal / n
[docs]def pdot(a, b):
"""Pairwise dot product.
``a`` must be the same shape as ``b``.
Parameters
----------
a: :class:`numpy.ndarray` of shape (N, M)
b: :class:`numpy.ndarray` of shape (N, M)
Returns
-------
:class:`numpy.ndarray` of shape (N,)
"""
return np.einsum('ij,ij->i', a, b)
[docs]def pnorm(a):
"""Euclidean norm of each vector in a matrix
Parameters
----------
a: :class:`numpy.ndarray` of shape (N, M)
Returns
-------
:class:`numpy.ndarray` of shape (N,)
"""
return pdot(a, a)**0.5
[docs]def angle(a, b):
"""Returns the angle between two vectors in radians
.. versionchanged:: 0.11.0
Moved into lib.mdamath
.. versionchanged:: 1.0.0
Changed rounding-off code to use `np.clip()`. Values lower than
-1.0 now return `np.pi` instead of `-np.pi`
"""
x = np.dot(a, b) / (norm(a) * norm(b))
# catch roundoffs that lead to nan otherwise
x = np.clip(x, -1.0, 1.0)
return np.arccos(x)
[docs]def stp(vec1, vec2, vec3):
r"""Takes the scalar triple product of three vectors.
Returns the volume *V* of the parallel epiped spanned by the three
vectors
.. math::
V = \mathbf{v}_3 \cdot (\mathbf{v}_1 \times \mathbf{v}_2)
.. versionchanged:: 0.11.0
Moved into lib.mdamath
"""
return np.dot(vec3, np.cross(vec1, vec2))
[docs]def dihedral(ab, bc, cd):
r"""Returns the dihedral angle in radians between vectors connecting A,B,C,D.
The dihedral measures the rotation around bc::
ab
A---->B
\ bc
_\'
C---->D
cd
The dihedral angle is restricted to the range -π <= x <= π.
.. versionadded:: 0.8
.. versionchanged:: 0.11.0
Moved into lib.mdamath
"""
x = angle(normal(ab, bc), normal(bc, cd))
return (x if stp(ab, bc, cd) <= 0.0 else -x)
[docs]def sarrus_det(matrix):
"""Computes the determinant of a 3x3 matrix according to the
`rule of Sarrus`_.
If an array of 3x3 matrices is supplied, determinants are computed per
matrix and returned as an appropriately shaped numpy array.
.. _rule of Sarrus:
https://en.wikipedia.org/wiki/Rule_of_Sarrus
Parameters
----------
matrix : numpy.ndarray
An array of shape ``(..., 3, 3)`` with the 3x3 matrices residing in the
last two dimensions.
Returns
-------
det : float or numpy.ndarray
The determinant(s) of `matrix`.
If ``matrix.shape == (3, 3)``, the determinant will be returned as a
scalar. If ``matrix.shape == (..., 3, 3)``, the determinants will be
returned as a :class:`numpy.ndarray` of shape ``(...,)`` and dtype
``numpy.float64``.
Raises
------
ValueError:
If `matrix` has less than two dimensions or its last two dimensions
are not of shape ``(3, 3)``.
.. versionadded:: 0.20.0
"""
m = matrix.astype(np.float64)
shape = m.shape
ndim = m.ndim
if ndim < 2 or shape[-2:] != (3, 3):
raise ValueError("Invalid matrix shape: must be (3, 3) or (..., 3, 3), "
"got {}.".format(shape))
if ndim == 2:
return _sarrus_det_single(m)
return _sarrus_det_multiple(m.reshape((-1, 3, 3))).reshape(shape[:-2])
[docs]def triclinic_box(x, y, z):
"""Convert the three triclinic box vectors to
``[lx, ly, lz, alpha, beta, gamma]``.
If the resulting box is invalid, i.e., any box length is zero or negative,
or any angle is outside the open interval ``(0, 180)``, a zero vector will
be returned.
All angles are in degrees and defined as follows:
* ``alpha = angle(y,z)``
* ``beta = angle(x,z)``
* ``gamma = angle(x,y)``
Parameters
----------
x : array_like
Array of shape ``(3,)`` representing the first box vector
y : array_like
Array of shape ``(3,)`` representing the second box vector
z : array_like
Array of shape ``(3,)`` representing the third box vector
Returns
-------
numpy.ndarray
A numpy array of shape ``(6,)`` and dtype ``np.float32`` providing the
unitcell dimensions in the same format as returned by
:attr:`MDAnalysis.coordinates.base.Timestep.dimensions`:\n
``[lx, ly, lz, alpha, beta, gamma]``.\n
Invalid boxes are returned as a zero vector.
Note
----
Definition of angles: http://en.wikipedia.org/wiki/Lattice_constant
See Also
--------
:func:`~MDAnalysis.lib.mdamath.triclinic_vectors`
.. versionchanged:: 0.20.0
Calculations are performed in double precision and invalid box vectors
result in an all-zero box.
"""
x = np.asarray(x, dtype=np.float64)
y = np.asarray(y, dtype=np.float64)
z = np.asarray(z, dtype=np.float64)
lx = norm(x)
ly = norm(y)
lz = norm(z)
alpha = np.rad2deg(np.arccos(np.dot(y, z) / (ly * lz)))
beta = np.rad2deg(np.arccos(np.dot(x, z) / (lx * lz)))
gamma = np.rad2deg(np.arccos(np.dot(x, y) / (lx * ly)))
box = np.array([lx, ly, lz, alpha, beta, gamma], dtype=np.float32)
# Only positive edge lengths and angles in (0, 180) are allowed:
if np.all(box > 0.0) and alpha < 180.0 and beta < 180.0 and gamma < 180.0:
return box
# invalid box, return zero vector:
return np.zeros(6, dtype=np.float32)
[docs]def triclinic_vectors(dimensions, dtype=np.float32):
"""Convert ``[lx, ly, lz, alpha, beta, gamma]`` to a triclinic matrix
representation.
Original `code by Tsjerk Wassenaar`_ posted on the Gromacs mailinglist.
If `dimensions` indicates a non-periodic system (i.e., all lengths are
zero), zero vectors are returned. The same applies for invalid `dimensions`,
i.e., any box length is zero or negative, or any angle is outside the open
interval ``(0, 180)``.
.. _code by Tsjerk Wassenaar:
http://www.mail-archive.com/[email protected]/msg28032.html
Parameters
----------
dimensions : array_like
Unitcell dimensions provided in the same format as returned by
:attr:`MDAnalysis.coordinates.base.Timestep.dimensions`:\n
``[lx, ly, lz, alpha, beta, gamma]``.
dtype: numpy.dtype
The data type of the returned box matrix.
Returns
-------
box_matrix : numpy.ndarray
A numpy array of shape ``(3, 3)`` and dtype `dtype`,
with ``box_matrix[0]`` containing the first, ``box_matrix[1]`` the
second, and ``box_matrix[2]`` the third box vector.
Notes
-----
* The first vector is guaranteed to point along the x-axis, i.e., it has the
form ``(lx, 0, 0)``.
* The second vector is guaranteed to lie in the x/y-plane, i.e., its
z-component is guaranteed to be zero.
* If any box length is negative or zero, or if any box angle is zero, the
box is treated as invalid and an all-zero-matrix is returned.
.. versionchanged:: 0.7.6
Null-vectors are returned for non-periodic (or missing) unit cell.
.. versionchanged:: 0.20.0
* Calculations are performed in double precision and zero vectors are
also returned for invalid boxes.
* Added optional output dtype parameter.
"""
dim = np.asarray(dimensions, dtype=np.float64)
lx, ly, lz, alpha, beta, gamma = dim
# Only positive edge lengths and angles in (0, 180) are allowed:
if not (np.all(dim > 0.0) and
alpha < 180.0 and beta < 180.0 and gamma < 180.0):
# invalid box, return zero vectors:
box_matrix = np.zeros((3, 3), dtype=dtype)
# detect orthogonal boxes:
elif alpha == beta == gamma == 90.0:
# box is orthogonal, return a diagonal matrix:
box_matrix = np.diag(dim[:3].astype(dtype, copy=False))
# we have a triclinic box:
else:
box_matrix = np.zeros((3, 3), dtype=np.float64)
box_matrix[0, 0] = lx
# Use exact trigonometric values for right angles:
if alpha == 90.0:
cos_alpha = 0.0
else:
cos_alpha = np.cos(np.deg2rad(alpha))
if beta == 90.0:
cos_beta = 0.0
else:
cos_beta = np.cos(np.deg2rad(beta))
if gamma == 90.0:
cos_gamma = 0.0
sin_gamma = 1.0
else:
gamma = np.deg2rad(gamma)
cos_gamma = np.cos(gamma)
sin_gamma = np.sin(gamma)
box_matrix[1, 0] = ly * cos_gamma
box_matrix[1, 1] = ly * sin_gamma
box_matrix[2, 0] = lz * cos_beta
box_matrix[2, 1] = lz * (cos_alpha - cos_beta * cos_gamma) / sin_gamma
box_matrix[2, 2] = np.sqrt(lz * lz - box_matrix[2, 0] ** 2 -
box_matrix[2, 1] ** 2)
# The discriminant of the above square root is only negative or zero for
# triplets of box angles that lead to an invalid box (i.e., the sum of
# any two angles is less than or equal to the third).
# We don't need to explicitly test for np.nan here since checking for a
# positive value already covers that.
if box_matrix[2, 2] > 0.0:
# all good, convert to correct dtype:
box_matrix = box_matrix.astype(dtype, copy=False)
else:
# invalid box, return zero vectors:
box_matrix = np.zeros((3, 3), dtype=dtype)
return box_matrix
[docs]def box_volume(dimensions):
"""Return the volume of the unitcell described by `dimensions`.
The volume is computed as the product of the box matrix trace, with the
matrix obtained from :func:`triclinic_vectors`.
If the box is invalid, i.e., any box length is zero or negative, or any
angle is outside the open interval ``(0, 180)``, the resulting volume will
be zero.
Parameters
----------
dimensions : array_like
Unitcell dimensions provided in the same format as returned by
:attr:`MDAnalysis.coordinates.base.Timestep.dimensions`:\n
``[lx, ly, lz, alpha, beta, gamma]``.
Returns
-------
volume : float
The volume of the unitcell. Will be zero for invalid boxes.
.. versionchanged:: 0.20.0
Calculations are performed in double precision and zero is returned
for invalid dimensions.
"""
dim = np.asarray(dimensions, dtype=np.float64)
lx, ly, lz, alpha, beta, gamma = dim
if alpha == beta == gamma == 90.0 and lx > 0 and ly > 0 and lz > 0:
# valid orthogonal box, volume is the product of edge lengths:
volume = lx * ly * lz
else:
# triclinic or invalid box, volume is trace product of box matrix
# (invalid boxes are set to all zeros by triclinic_vectors):
tri_vecs = triclinic_vectors(dim, dtype=np.float64)
volume = tri_vecs[0, 0] * tri_vecs[1, 1] * tri_vecs[2, 2]
return volume