hikari.symmetry.operations

This file contains class and definitions of 3-dimensional symmetry operations.

Classes

`Operation`	Class storing information about symmetry operations, with clear string
`BoundedOperation`	A subclass of Operation where all three elements of the translation
`PointOperation`	A subclass of BoundedOperation, asserts translation vector = [0,0,0]

Module Contents

class hikari.symmetry.operations.Operation(transformation, translation=np.array([0, 0, 0]))[source]

Class storing information about symmetry operations, with clear string representation and intuitive syntax for combining / utilising operations: “=” to compare two symmetry operations for logical equivalence, “*” to combine two symmetry operations or transform a vector, “** n” to apply symmetry operation n times, “% n” to restrict symmetry operation to n unit cells. Some of the functions may work incorrectly for rhombohedral unit cells #TODO

class Type[source]

Bases: enum.Enum

Enumerator class storing information about type of operation

rotoinversion = 4

identity = 3

reflection = 2

rotation = 1

inversion = 0

rototranslation = -1

transflection = -2

translation = -3

property tf: numpy.ndarray

Return type:: numpy.ndarray

property tl: numpy.ndarray

Return type:: numpy.ndarray

__eq__(other)[source]

Parameters:: other (Operation)
Return type:: bool

__mul__(other)[source]

Parameters:: other (Operation)
Return type:: Operation

__pow__(power, modulo=None)[source]

Parameters:: power (int)
Return type:: Operation

__repr__()[source]

Return type:: str

__str__()[source]

Return type:: str

__hash__()[source]

Return type:: int

classmethod from_code(code)[source]

Create new symmetry operation using symmetry code “x’,y’,z’” containing transformation of each individual coordinate from x,y,z to x’,y’,z’.

Parameters:: code (str) – string representing new coordinates after operation
Returns:: Symmetry operation generated from given coordinate triplet code
Return type:: Operation

classmethod from_matrix(matrix)[source]

Create new symmetry operation using augmented 4x4 transformation matrix

Parameters:: matrix (numpy.ndarray) – augmented 4x4 matrix
Returns:: Symmetry operation generated based on augmented matrix
Return type:: Operation

classmethod from_pair(matrix, vector)[source]

Create new symmetry operation using point transformation 3x3 matrix and 3-length translation vector. Alias for standard creation method.

Parameters:

matrix (numpy.ndarray) – 3x3 point transformation matrix
vector (numpy.ndarray) – 3-length translation vector

Returns:

Symmetry operation generated based on matrix - vector pair

Return type:

Operation

static _row_to_str(xyz, r)[source]

Convert xyz: 3-el. list and r - number to single element of code triplet

Parameters:

xyz (numpy.ndarray[int]) – 3-element list of coordinates, e.g.: [1,-1,0]
r (float) – translation applied to the row

Returns:

string representing change of coordinate

Return type:

str

static _project(vector, onto)[source]

Return projection of np.ndarray “vector” to np.ndarray “onto”

Parameters:

vector (numpy.ndarray)
onto (numpy.ndarray)

Return type:

numpy.ndarray

property code: str

Return type:: str

property matrix: numpy.ndarray

Augmented 4 x 4 transformation matrix with float-type values

Return type:: numpy.ndarray

property det: int

Determinant of 3x3 transformation part of operation’s matrix

Return type:: int

property typ: Type

Crystallographic type of this symmetry operation (see Type)

Return type:: Type

property name: str

‘m’, ‘3’ or ‘2_1’

Type:: Short name of symmetry operation, e.g.
Return type:: str

property fold: int

Number of times operation must be repeated to become identity, inversion or translation: n for n-fold axes, 2 for reflections, 1 for other (max 6)

Return type:: int

property order: int

Number of times operation has to be repeated to become a translation, e.g.: n for all n-fold axes, 2 for other (max 6)

Return type:: int

property glide: numpy.ndarray

Part of the translation vector stemming from operations’ glide

Return type:: numpy.ndarray

property glide_fold: int

Number of types glide component of the operation must be repeated in order to contain only integer values, e.g.: 3 for “6_2”, 4 for “d”

Return type:: int

property origin: numpy.ndarray

Selected point that remains on the symmetry element after operation Loosely based on solving https://math.stackexchange.com/q/1054481/.

Return type:: numpy.ndarray

property reciprocal: PointOperation

Relevant symmetry operation in its respective reciprocal space

Return type:: PointOperation

property trace: int

Trace of 3x3 transformation part of operation’s matrix

Return type:: int

property translational: bool

True if operation has any glide component, False otherwise

Return type:: bool

property invariants: list[numpy.ndarray]

List of directions not affected by this symmetry operation

Return type:: list[numpy.ndarray]

property orientation: numpy.ndarray | None

Direction of symmetry element (if it can be defined) else None

Return type:: Union[numpy.ndarray, None]

property sense: str

“+” or “-”, the “sense” of rotation, as given in ITC A, 11.1.2

Return type:: str

property _hm_span

property _hm_glide

property hm_symbol: str

Return type:: str

property bounded: BoundedOperation

Instance of self that always collapses down to Coset representative

Return type:: BoundedOperation

property unbounded: Operation

Instance of self that can express translations beyond the unit cube

Return type:: Operation

property is_bounded: bool

True if self is a coset representatives i.e. 0 <= tl < 1 element-wise

Return type:: bool

at(point)[source]

Transform operation as little as possible so that its symmetry element contains “point”. To be used after “into” if used together. Based on “ITC 5.2.1. Transformations”.

Parameters:: point (numpy.ndarray) – Target coordinates of point which should lie in element
Returns:: New symmetry operation which contains “point” in its element
Return type:: Operation

into(direction, hexagonal=False)[source]

Rotate operation so that its orientation changes to “direction”, while preserving fractional glide. To be used before respective “at” method. Will most likely not work for unimplemented rhombohedral unit cells.

Parameters:

direction (numpy.ndarray) – Target orientation for element of symmetry operation
hexagonal (bool) – True if operation is defined in hexagonal coordinates

Returns:

New symmetry operation whose orientation is “direction”

Return type:

Operation

transform(other)[source]

Transform a column containing rows of coordinate points

Parameters:: other (numpy.ndarray) – A vertical numpy array of coordinate triplets kept in rows
Returns:: Same-shaped array of coordinate triplets transformed by self
Return type:: numpy.ndarray

extincts(hkl)[source]

Return boolean array with truth whenever reflection should be extinct

Parameters:: hkl (numpy.ndarray) – An array containing one hkl or multiple hkls in columns
Returns:: array of booleans, where extinct reflections are marked as True
Return type:: numpy.ndarray

distance_to_point(point)[source]

Parameters:: point (Sequence[Union[int, float]])
Return type:: float

class hikari.symmetry.operations.BoundedOperation(transformation, translation=np.array([0, 0, 0]))[source]

Bases: Operation

A subclass of Operation where all three elements of the translation vector are bounded between 0 (inclusive) and 1 (exclusive). This class is suitable for handling coset representatives of space groups, since upon binding operations related by unit translation become equivalent.

property tl: numpy.ndarray

Return type:: numpy.ndarray

class hikari.symmetry.operations.PointOperation(transformation, translation=np.array([0, 0, 0]))[source]

Bases: BoundedOperation

A subclass of BoundedOperation, asserts translation vector = [0,0,0]

property tl: numpy.ndarray

Return type:: numpy.ndarray