Source code for slothpy._general_utilities._grids_over_hemisphere
# SlothPy
# Copyright (C) 2023 Mikolaj Tadeusz Zychowicz (MTZ)
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
from numpy import ndarray, array
[docs]
def lebedev_laikov_grid(grid: int) -> ndarray:
"""
Returns Lebedev-Laikov-Grids [1] over a hemisphere.
Parameters
----------
grid : int
An integer from 0 to 11 controling the number of points in the grid
over the hemisphere.
0 - 17 points,
1 - 29 points,
2 - 93 points,
3 - 161 points,
4 - 185 points,
5 - 229 points,
6 - 309 points,
7 - 401 points,
8 - 505 points,
9 - 889 points,
10 - 1381 points,
11 - 2949 points
Returns
-------
ndarray
Grid array in the form [[direction_x, direction_y, direction_z,
weight], ...].
Raises
------
ValueError
If grid is not an integer from 0 to 11.
Notes
-----
Grids 0-2 can be considered as insufficient in most cases (only should be
used for experimental purposes), 3-5 minimal to standard, 6-8 very precise,
but costly, 9-11 very dense - only for particular use-cases (extremely
costly - especially 10-11).
References
----------
.. [1] V.I. Lebedev, and D.N. Laikov
"A quadrature formula for the sphere of the 131st
algebraic order of accuracy"
Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 477-481.
"""
if grid == 0:
grid_array = array(
[
[1.0, 0.0, 0.0, 0.047619047619],
[-1.0, 0.0, 0.0, 0.047619047619],
[0.0, 1.0, 0.0, 0.047619047619],
[0.0, -1.0, 0.0, 0.047619047619],
[0.0, 0.0, 1.0, 0.095238095238],
[0.0, 0.707106781187, 0.707106781187, 0.07619047619],
[0.0, -0.707106781187, 0.707106781187, 0.07619047619],
[0.707106781187, 0.0, 0.707106781187, 0.07619047619],
[-0.707106781187, 0.0, 0.707106781187, 0.07619047619],
[0.707106781187, 0.707106781187, 0.0, 0.038095238095],
[-0.707106781187, 0.707106781187, 0.0, 0.038095238095],
[0.707106781187, -0.707106781187, 0.0, 0.038095238095],
[-0.707106781187, -0.707106781187, 0.0, 0.038095238095],
[0.57735026919, 0.57735026919, 0.57735026919, 0.064285714286],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.064285714286],
[0.57735026919, -0.57735026919, 0.57735026919, 0.064285714286],
[
-0.57735026919,
-0.57735026919,
0.57735026919,
0.064285714286,
],
]
)
return grid_array
if grid == 1:
grid_array = array(
[
[1.0, 0.0, 0.0, 0.012698412698],
[-1.0, 0.0, 0.0, 0.012698412698],
[0.0, 1.0, 0.0, 0.012698412698],
[0.0, -1.0, 0.0, 0.012698412698],
[0.0, 0.0, 1.0, 0.025396825397],
[0.0, 0.707106781187, 0.707106781187, 0.045149911817],
[0.0, -0.707106781187, 0.707106781187, 0.045149911817],
[0.707106781187, 0.0, 0.707106781187, 0.045149911817],
[-0.707106781187, 0.0, 0.707106781187, 0.045149911817],
[0.707106781187, 0.707106781187, 0.0, 0.022574955908],
[-0.707106781187, 0.707106781187, 0.0, 0.022574955908],
[0.707106781187, -0.707106781187, 0.0, 0.022574955908],
[-0.707106781187, -0.707106781187, 0.0, 0.022574955908],
[0.57735026919, 0.57735026919, 0.57735026919, 0.0421875],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.0421875],
[0.57735026919, -0.57735026919, 0.57735026919, 0.0421875],
[-0.57735026919, -0.57735026919, 0.57735026919, 0.0421875],
[
0.301511344578,
0.301511344578,
0.904534033733,
0.040346671076,
],
[
-0.301511344578,
0.301511344578,
0.904534033733,
0.040346671076,
],
[
0.301511344578,
-0.301511344578,
0.904534033733,
0.040346671076,
],
[
-0.301511344578,
-0.301511344578,
0.904534033733,
0.040346671076,
],
[
0.301511344578,
0.904534033733,
0.301511344578,
0.040346671076,
],
[
-0.301511344578,
0.904534033733,
0.301511344578,
0.040346671076,
],
[
0.301511344578,
-0.904534033733,
0.301511344578,
0.040346671076,
],
[
-0.301511344578,
-0.904534033733,
0.301511344578,
0.040346671076,
],
[
0.904534033733,
0.301511344578,
0.301511344578,
0.040346671076,
],
[
-0.904534033733,
0.301511344578,
0.301511344578,
0.040346671076,
],
[
0.904534033733,
-0.301511344578,
0.301511344578,
0.040346671076,
],
[
-0.904534033733,
-0.301511344578,
0.301511344578,
0.040346671076,
],
]
)
return grid_array
if grid == 2:
grid_array = array(
[
[1.0, 0.0, 0.0, 0.005544842902],
[-1.0, 0.0, 0.0, 0.005544842902],
[0.0, 1.0, 0.0, 0.005544842902],
[0.0, -1.0, 0.0, 0.005544842902],
[0.0, 0.0, 1.0, 0.011089685804],
[0.0, 0.707106781187, 0.707106781187, 0.012142665541],
[0.0, -0.707106781187, 0.707106781187, 0.012142665541],
[0.707106781187, 0.0, 0.707106781187, 0.012142665541],
[-0.707106781187, 0.0, 0.707106781187, 0.012142665541],
[0.707106781187, 0.707106781187, 0.0, 0.006071332771],
[-0.707106781187, 0.707106781187, 0.0, 0.006071332771],
[0.707106781187, -0.707106781187, 0.0, 0.006071332771],
[-0.707106781187, -0.707106781187, 0.0, 0.006071332771],
[0.57735026919, 0.57735026919, 0.57735026919, 0.012767349547],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.012767349547],
[0.57735026919, -0.57735026919, 0.57735026919, 0.012767349547],
[
-0.57735026919,
-0.57735026919,
0.57735026919,
0.012767349547,
],
[
0.255125262111,
0.255125262111,
0.932642590313,
0.010366775175,
],
[
-0.255125262111,
0.255125262111,
0.932642590313,
0.010366775175,
],
[
0.255125262111,
-0.255125262111,
0.932642590313,
0.010366775175,
],
[
-0.255125262111,
-0.255125262111,
0.932642590313,
0.010366775175,
],
[
0.255125262111,
0.932642590313,
0.255125262111,
0.010366775175,
],
[
-0.255125262111,
0.932642590313,
0.255125262111,
0.010366775175,
],
[
0.255125262111,
-0.932642590313,
0.255125262111,
0.010366775175,
],
[
-0.255125262111,
-0.932642590313,
0.255125262111,
0.010366775175,
],
[
0.932642590313,
0.255125262111,
0.255125262111,
0.010366775175,
],
[
-0.932642590313,
0.255125262111,
0.255125262111,
0.010366775175,
],
[
0.932642590313,
-0.255125262111,
0.255125262111,
0.010366775175,
],
[
-0.932642590313,
-0.255125262111,
0.255125262111,
0.010366775175,
],
[
0.674360146036,
0.674360146036,
0.300793595138,
0.01263585802,
],
[
-0.674360146036,
0.674360146036,
0.300793595138,
0.01263585802,
],
[
0.674360146036,
-0.674360146036,
0.300793595138,
0.01263585802,
],
[
-0.674360146036,
-0.674360146036,
0.300793595138,
0.01263585802,
],
[
0.674360146036,
0.300793595138,
0.674360146036,
0.01263585802,
],
[
-0.674360146036,
0.300793595138,
0.674360146036,
0.01263585802,
],
[
0.674360146036,
-0.300793595138,
0.674360146036,
0.01263585802,
],
[
-0.674360146036,
-0.300793595138,
0.674360146036,
0.01263585802,
],
[
0.300793595138,
0.674360146036,
0.674360146036,
0.01263585802,
],
[
-0.300793595138,
0.674360146036,
0.674360146036,
0.01263585802,
],
[
0.300793595138,
-0.674360146036,
0.674360146036,
0.01263585802,
],
[
-0.300793595138,
-0.674360146036,
0.674360146036,
0.01263585802,
],
[
0.431891069672,
0.431891069672,
0.791795559393,
0.012403340013,
],
[
-0.431891069672,
0.431891069672,
0.791795559393,
0.012403340013,
],
[
0.431891069672,
-0.431891069672,
0.791795559393,
0.012403340013,
],
[
-0.431891069672,
-0.431891069672,
0.791795559393,
0.012403340013,
],
[
0.431891069672,
0.791795559393,
0.431891069672,
0.012403340013,
],
[
-0.431891069672,
0.791795559393,
0.431891069672,
0.012403340013,
],
[
0.431891069672,
-0.791795559393,
0.431891069672,
0.012403340013,
],
[
-0.431891069672,
-0.791795559393,
0.431891069672,
0.012403340013,
],
[
0.791795559393,
0.431891069672,
0.431891069672,
0.012403340013,
],
[
-0.791795559393,
0.431891069672,
0.431891069672,
0.012403340013,
],
[
0.791795559393,
-0.431891069672,
0.431891069672,
0.012403340013,
],
[
-0.791795559393,
-0.431891069672,
0.431891069672,
0.012403340013,
],
[0.261393136034, 0.965232421976, 0.0, 0.005477143385],
[-0.261393136034, 0.965232421976, 0.0, 0.005477143385],
[0.261393136034, -0.965232421976, 0.0, 0.005477143385],
[-0.261393136034, -0.965232421976, 0.0, 0.005477143385],
[0.965232421976, 0.261393136034, 0.0, 0.005477143385],
[-0.965232421976, 0.261393136034, 0.0, 0.005477143385],
[0.965232421976, -0.261393136034, 0.0, 0.005477143385],
[-0.965232421976, -0.261393136034, 0.0, 0.005477143385],
[0.261393136034, 0.0, 0.965232421976, 0.01095428677],
[-0.261393136034, 0.0, 0.965232421976, 0.01095428677],
[0.965232421976, 0.0, 0.261393136034, 0.01095428677],
[-0.965232421976, 0.0, 0.261393136034, 0.01095428677],
[0.0, 0.261393136034, 0.965232421976, 0.01095428677],
[0.0, -0.261393136034, 0.965232421976, 0.01095428677],
[0.0, 0.965232421976, 0.261393136034, 0.01095428677],
[0.0, -0.965232421976, 0.261393136034, 0.01095428677],
[
0.49904531618,
0.144663074433,
0.854415804685,
0.011936767975,
],
[
-0.49904531618,
0.144663074433,
0.854415804685,
0.011936767975,
],
[
0.49904531618,
-0.144663074433,
0.854415804685,
0.011936767975,
],
[
-0.49904531618,
-0.144663074433,
0.854415804685,
0.011936767975,
],
[
0.49904531618,
0.854415804685,
0.144663074433,
0.011936767975,
],
[
-0.49904531618,
0.854415804685,
0.144663074433,
0.011936767975,
],
[
0.49904531618,
-0.854415804685,
0.144663074433,
0.011936767975,
],
[
-0.49904531618,
-0.854415804685,
0.144663074433,
0.011936767975,
],
[
0.144663074433,
0.49904531618,
0.854415804685,
0.011936767975,
],
[
-0.144663074433,
0.49904531618,
0.854415804685,
0.011936767975,
],
[
0.144663074433,
-0.49904531618,
0.854415804685,
0.011936767975,
],
[
-0.144663074433,
-0.49904531618,
0.854415804685,
0.011936767975,
],
[
0.144663074433,
0.854415804685,
0.49904531618,
0.011936767975,
],
[
-0.144663074433,
0.854415804685,
0.49904531618,
0.011936767975,
],
[
0.144663074433,
-0.854415804685,
0.49904531618,
0.011936767975,
],
[
-0.144663074433,
-0.854415804685,
0.49904531618,
0.011936767975,
],
[
0.854415804685,
0.49904531618,
0.144663074433,
0.011936767975,
],
[
-0.854415804685,
0.49904531618,
0.144663074433,
0.011936767975,
],
[
0.854415804685,
-0.49904531618,
0.144663074433,
0.011936767975,
],
[
-0.854415804685,
-0.49904531618,
0.144663074433,
0.011936767975,
],
[
0.854415804685,
0.144663074433,
0.49904531618,
0.011936767975,
],
[
-0.854415804685,
0.144663074433,
0.49904531618,
0.011936767975,
],
[
0.854415804685,
-0.144663074433,
0.49904531618,
0.011936767975,
],
[
-0.854415804685,
-0.144663074433,
0.49904531618,
0.011936767975,
],
]
)
return grid_array
if grid == 3:
grid_array = array(
[
[1.0, 0.0, 0.0, 0.000854591173],
[-1.0, 0.0, 0.0, 0.000854591173],
[0.0, 1.0, 0.0, 0.000854591173],
[0.0, -1.0, 0.0, 0.000854591173],
[0.0, 0.0, 1.0, 0.001709182345],
[0.57735026919, 0.57735026919, 0.57735026919, 0.00719823857],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.00719823857],
[0.57735026919, -0.57735026919, 0.57735026919, 0.00719823857],
[-0.57735026919, -0.57735026919, 0.57735026919, 0.00719823857],
[
0.351564034557,
0.351564034557,
0.867643624544,
0.006899576849,
],
[
-0.351564034557,
0.351564034557,
0.867643624544,
0.006899576849,
],
[
0.351564034557,
-0.351564034557,
0.867643624544,
0.006899576849,
],
[
-0.351564034557,
-0.351564034557,
0.867643624544,
0.006899576849,
],
[
0.351564034557,
0.867643624544,
0.351564034557,
0.006899576849,
],
[
-0.351564034557,
0.867643624544,
0.351564034557,
0.006899576849,
],
[
0.351564034557,
-0.867643624544,
0.351564034557,
0.006899576849,
],
[
-0.351564034557,
-0.867643624544,
0.351564034557,
0.006899576849,
],
[
0.867643624544,
0.351564034557,
0.351564034557,
0.006899576849,
],
[
-0.867643624544,
0.351564034557,
0.351564034557,
0.006899576849,
],
[
0.867643624544,
-0.351564034557,
0.351564034557,
0.006899576849,
],
[
-0.867643624544,
-0.351564034557,
0.351564034557,
0.006899576849,
],
[
0.656632941022,
0.656632941022,
0.371034178385,
0.007209645203,
],
[
-0.656632941022,
0.656632941022,
0.371034178385,
0.007209645203,
],
[
0.656632941022,
-0.656632941022,
0.371034178385,
0.007209645203,
],
[
-0.656632941022,
-0.656632941022,
0.371034178385,
0.007209645203,
],
[
0.656632941022,
0.371034178385,
0.656632941022,
0.007209645203,
],
[
-0.656632941022,
0.371034178385,
0.656632941022,
0.007209645203,
],
[
0.656632941022,
-0.371034178385,
0.656632941022,
0.007209645203,
],
[
-0.656632941022,
-0.371034178385,
0.656632941022,
0.007209645203,
],
[
0.371034178385,
0.656632941022,
0.656632941022,
0.007209645203,
],
[
-0.371034178385,
0.656632941022,
0.656632941022,
0.007209645203,
],
[
0.371034178385,
-0.656632941022,
0.656632941022,
0.007209645203,
],
[
-0.371034178385,
-0.656632941022,
0.656632941022,
0.007209645203,
],
[
0.472905413258,
0.472905413258,
0.743452042988,
0.007153459323,
],
[
-0.472905413258,
0.472905413258,
0.743452042988,
0.007153459323,
],
[
0.472905413258,
-0.472905413258,
0.743452042988,
0.007153459323,
],
[
-0.472905413258,
-0.472905413258,
0.743452042988,
0.007153459323,
],
[
0.472905413258,
0.743452042988,
0.472905413258,
0.007153459323,
],
[
-0.472905413258,
0.743452042988,
0.472905413258,
0.007153459323,
],
[
0.472905413258,
-0.743452042988,
0.472905413258,
0.007153459323,
],
[
-0.472905413258,
-0.743452042988,
0.472905413258,
0.007153459323,
],
[
0.743452042988,
0.472905413258,
0.472905413258,
0.007153459323,
],
[
-0.743452042988,
0.472905413258,
0.472905413258,
0.007153459323,
],
[
0.743452042988,
-0.472905413258,
0.472905413258,
0.007153459323,
],
[
-0.743452042988,
-0.472905413258,
0.472905413258,
0.007153459323,
],
[
0.096183085226,
0.096183085226,
0.990705621379,
0.004704202827,
],
[
-0.096183085226,
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)
return grid_array
if grid == 4:
grid_array = array(
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[
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)
return grid_array
if grid == 5:
grid_array = array(
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[0.707106781187, -0.707106781187, 0.0, 0.002548219972],
[-0.707106781187, -0.707106781187, 0.0, 0.002548219972],
[0.57735026919, 0.57735026919, 0.57735026919, 0.005024634838],
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[0.57735026919, -0.57735026919, 0.57735026919, 0.005024634838],
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]
)
return grid_array
if grid == 6:
grid_array = array(
[
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[-1.0, 0.0, 0.0, 0.00030951213],
[0.0, 1.0, 0.0, 0.00030951213],
[0.0, -1.0, 0.0, 0.00030951213],
[0.0, 0.0, 1.0, 0.000619024259],
[0.57735026919, 0.57735026919, 0.57735026919, 0.003704759397],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.003704759397],
[0.57735026919, -0.57735026919, 0.57735026919, 0.003704759397],
[
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],
[
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],
[
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[
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],
[
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],
[
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],
[
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],
[
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],
[
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[
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[
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[
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],
[
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],
[
-0.270356088359,
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)
return grid_array
if grid == 7:
grid_array = array(
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[0.0, -1.0, 0.0, 0.000219294209],
[0.0, 0.0, 1.0, 0.000438588418],
[0.0, 0.707106781187, 0.707106781187, 0.002872867235],
[0.0, -0.707106781187, 0.707106781187, 0.002872867235],
[0.707106781187, 0.0, 0.707106781187, 0.002872867235],
[-0.707106781187, 0.0, 0.707106781187, 0.002872867235],
[0.707106781187, 0.707106781187, 0.0, 0.001436433617],
[-0.707106781187, 0.707106781187, 0.0, 0.001436433617],
[0.707106781187, -0.707106781187, 0.0, 0.001436433617],
[-0.707106781187, -0.707106781187, 0.0, 0.001436433617],
[0.57735026919, 0.57735026919, 0.57735026919, 0.002843880689],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.002843880689],
[0.57735026919, -0.57735026919, 0.57735026919, 0.002843880689],
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[
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[
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)
return grid_array
if grid == 8:
grid_array = array(
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[0.0, -1.0, 0.0, 0.000143829419],
[0.0, 0.0, 1.0, 0.000287658838],
[0.57735026919, 0.57735026919, 0.57735026919, 0.002251544577],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.002251544577],
[0.57735026919, -0.57735026919, 0.57735026919, 0.002251544577],
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[
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[
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)
return grid_array
if grid == 9:
grid_array = array(
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[0.707106781187, -0.707106781187, 0.0, 0.000639828771],
[-0.707106781187, -0.707106781187, 0.0, 0.000639828771],
[0.57735026919, 0.57735026919, 0.57735026919, 0.001271437015],
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)
return grid_array
if grid == 10:
grid_array = array(
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[0.0, -1.0, 0.0, 2.9986751e-05],
[0.0, 0.0, 1.0, 5.9973503e-05],
[0.57735026919, 0.57735026919, 0.57735026919, 0.000815572106],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.000815572106],
[0.57735026919, -0.57735026919, 0.57735026919, 0.000815572106],
[
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[0.05250918173, 0.05250918173, 0.997238974202, 0.000382681729],
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],
[0.08993480082, 0.08993480082, 0.991878754285, 0.000490577315],
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[
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],
[
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[
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[
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[
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[
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)
return grid_array
if grid == 11:
grid_array = array(
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[-1.0, 0.0, 0.0, 9.735348e-06],
[0.0, 1.0, 0.0, 9.735348e-06],
[0.0, -1.0, 0.0, 9.735348e-06],
[0.0, 0.0, 1.0, 1.9470696e-05],
[0.0, 0.707106781187, 0.707106781187, 0.000381516248],
[0.0, -0.707106781187, 0.707106781187, 0.000381516248],
[0.707106781187, 0.0, 0.707106781187, 0.000381516248],
[-0.707106781187, 0.0, 0.707106781187, 0.000381516248],
[0.707106781187, 0.707106781187, 0.0, 0.000190758124],
[-0.707106781187, 0.707106781187, 0.0, 0.000190758124],
[0.707106781187, -0.707106781187, 0.0, 0.000190758124],
[-0.707106781187, -0.707106781187, 0.0, 0.000190758124],
[0.57735026919, 0.57735026919, 0.57735026919, 0.000380211909],
[-0.57735026919, 0.57735026919, 0.57735026919, 0.000380211909],
[0.57735026919, -0.57735026919, 0.57735026919, 0.000380211909],
[
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],
[
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],
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[
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[
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[
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],
[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
-0.761503592094,
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0.000379886928,
],
[
0.674725858837,
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[
-0.674725858837,
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[
0.674725858837,
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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[
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],
[
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0.000380904171,
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[
-0.73287487513,
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],
[0.67721357504, 0.029199461358, 0.735206886011, 0.0003811069],
[-0.67721357504, 0.029199461358, 0.735206886011, 0.0003811069],
[0.67721357504, -0.029199461358, 0.735206886011, 0.0003811069],
[
-0.67721357504,
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],
[0.67721357504, 0.735206886011, 0.029199461358, 0.0003811069],
[-0.67721357504, 0.735206886011, 0.029199461358, 0.0003811069],
[0.67721357504, -0.735206886011, 0.029199461358, 0.0003811069],
[
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0.029199461358,
0.0003811069,
],
[0.029199461358, 0.67721357504, 0.735206886011, 0.0003811069],
[-0.029199461358, 0.67721357504, 0.735206886011, 0.0003811069],
[0.029199461358, -0.67721357504, 0.735206886011, 0.0003811069],
[
-0.029199461358,
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0.735206886011,
0.0003811069,
],
[0.029199461358, 0.735206886011, 0.67721357504, 0.0003811069],
[-0.029199461358, 0.735206886011, 0.67721357504, 0.0003811069],
[0.029199461358, -0.735206886011, 0.67721357504, 0.0003811069],
[
-0.029199461358,
-0.735206886011,
0.67721357504,
0.0003811069,
],
[0.735206886011, 0.67721357504, 0.029199461358, 0.0003811069],
[-0.735206886011, 0.67721357504, 0.029199461358, 0.0003811069],
[0.735206886011, -0.67721357504, 0.029199461358, 0.0003811069],
[
-0.735206886011,
-0.67721357504,
0.029199461358,
0.0003811069,
],
[0.735206886011, 0.029199461358, 0.67721357504, 0.0003811069],
[-0.735206886011, 0.029199461358, 0.67721357504, 0.0003811069],
[0.735206886011, -0.029199461358, 0.67721357504, 0.0003811069],
[
-0.735206886011,
-0.029199461358,
0.67721357504,
0.0003811069,
],
]
)
return grid_array
raise ValueError(f"Grid number has to lay between 0-11")