Geometry#

For the full reference see geometry.

Import#

>>> # Exact import
>>> from radtools.geometry import volume
>>> # Recommended import
>>> from radtools import volume

Volume#

Volume can be computed from a several types of input data:

  • Cell

\(3\times3\) array, rows are the cell vectors, columns are the \(xyz\) coordinates.

>>> cell = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> volume(cell)
1.0

  • Three vectors

>>> a = [1, 0, 0]
>>> b = [0, 1, 0]
>>> c = [0, 0, 1]
>>> volume(a, b, c)
1.0

  • Six cell parameters

>>> a = 1
>>> b = 1
>>> c = 1
>>> alpha = 90
>>> beta = 90
>>> gamma = 90
>>> volume(a, b, c, alpha, beta, gamma)
1.0

Angle#

Computes the smallest angle between two vectors.

>>> from radtools import angle
>>> a = [1, 0, 0]
>>> b = [0, 1, 0]
>>> # In degrees by default
>>> angle(a, b)
90.0
>>> # In radians
>>> round(angle(a, b, radians=True), 4)
1.5708
>>> # For the zero vector the angle is not defined
>>> angle(a, [0, 0, 0])
Traceback (most recent call last):
...
ValueError: Angle is ill defined (zero vector).

Coordinates#

Computes relative coordinates of a vector with respect to some basis. Basis is defined by three vectors, which are passed to the function in a form of a \(3\times3\) array. Rows of the cell are the basis vectors, columns are the \(xyz\) coordinates.

>>> from radtools import absolute_to_relative
>>> basis = [[2, 0, 0], [0, 4, 0], [0, 0, 8]]
>>> vector = [1, 1, 1]
>>> absolute_to_relative(basis, vector)
array([0.5  , 0.25 , 0.125])

Parallelepiped#

Check if the parallelepiped can be formed from six parameters. \(a\), \(b\), \(c\) are the lengths of the three vectors forming the parallelepiped. \(\alpha\), \(\beta\), \(\gamma\) are the angles between the vectors.

>>> from radtools import parallelepiped_check
>>> a = 1
>>> b = 1
>>> c = 1
>>> alpha = 90
>>> beta = 90
>>> gamma = 90
>>> parallelepiped_check(a, b, c, alpha, beta, gamma)
True
>>> parallelepiped_check(a, b, c, alpha, beta, 181)
False

Orthonormal basis#

Spans an orthonormal basis from one vector. The input vector is normalized and forms \(\vec{e}_3\). The orthonormal basis is computed by rotation of the standard basis

\[\begin{split}\begin{aligned} \vec{e}_1^{st} &= (1, 0, 0) \\ \vec{e}_2^{st} &= (0, 1, 0) \\ \vec{e}_3^{st} &= (0, 0, 1) \\ \end{aligned}\end{split}\]

with the rotation around the axis perpendicular to both \(\vec{e}_3^{st}\) and \(\vec{e}_3\) by the angle \(\theta\) between them.

>>> from radtools import span_orthonormal_set
>>> vector = [1, 1, 1]
>>> span_orthonormal_set(vector)
array([[ 0.78867513, -0.21132487, -0.57735027],
       [-0.21132487,  0.78867513, -0.57735027],
       [ 0.57735027,  0.57735027,  0.57735027]])