qpms/lattices2d.py

   1 import numpy as np
   2 from enum import Enum
   3 from math import floor
   4
   5 nx = None
   6
   7 class LatticeType(Enum):
   8     """
   9     All the five Bravais lattices in 2D
  10     """
  11     OBLIQUE=1
  12     RECTANGULAR=2
  13     SQUARE=4
  14     RHOMBIC=5
  15     EQUILATERAL_TRIANGULAR=3
  16     RIGHT_ISOSCELES=SQUARE
  17     PARALLELOGRAMMIC=OBLIQUE
  18     CENTERED_RHOMBIC=RECTANGULAR
  19     RIGHT_TRIANGULAR=RECTANGULAR
  20     CENTERED_RECTANGULAR=RHOMBIC
  21     ISOSCELE_TRIANGULAR=RHOMBIC
  22     RIGHT_ISOSCELE_TRIANGULAR=SQUARE
  23     HEXAGONAL=EQUILATERAL_TRIANGULAR
  24
  25 def reduceBasisSingle(b1, b2):
  26     """
  27     Lagrange-Gauss reduction of a 2D basis.
  28     cf. https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch17.pdf
  29     inputs and outputs are (2,)-shaped numpy arrays
  30     The output shall satisfy |b1| <= |b2| <= |b2 - b1|
  31     TODO doc
  32
  33     TODO perhaps have the (on-demand?) guarantee of obtuse angle between b1, b2?
  34     TODO possibility of returning the (in-order, no-obtuse angles) b as well?
  35     """
  36     b1 = np.array(b1)
  37     b2 = np.array(b2)
  38     if b1.shape != (2,) or b2.shape != (2,):
  39         raise ValueError('Shape of b1 and b2 must be (2,)')
  40     B1 = np.sum(b1 * b1, axis=-1, keepdims=True)
  41     mu = np.sum(b1 * b2, axis=-1, keepdims=True) / B1
  42     b2 = b2 - np.rint(mu) * b1
  43     B2 = np.sum(b2 * b2, axis=-1, keepdims=True)
  44     while(np.any(B2 < B1)):
  45         b2t = b1
  46         b1 = b2
  47         b2 = b2t
  48         B1 = B2
  49         mu = np.sum(b1 * b2, axis=-1, keepdims=True) / B1
  50         b2 = b2 - np.rint(mu) * b1
  51         B2 = np.sum(b2*b2, axis=-1, keepdims=True)
  52     return np.array((b1,b2))
  53
  54 def shortestBase3(b1, b2):
  55     '''
  56     returns the "ordered shortest triple" of base vectors (each pair from
  57     the triple is a base) and there may not be obtuse angle between b1, b2
  58     and between b2, b3
  59     '''
  60     b1, b2 = reduceBasisSingle(b1,b2)
  61     if is_obtuse(b1, b2, tolerance=0):
  62         b3 = b2
  63         b2 = b2 + b1
  64     else:
  65         b3 = b2 - b1
  66     return (b1, b2, b3)
  67
  68 def shortestBase46(b1, b2, tolerance=1e-13):
  69     b1, b2 = reduceBasisSingle(b1,b2)
  70     b1s = np.sum(b1 ** 2)
  71     b2s = np.sum(b2 ** 2)
  72     b3 = b2 - b1
  73     b3s = np.sum(b3 ** 2)
  74     eps = tolerance * (b2s + b1s)
  75     if abs(b3s - b2s - b1s) < eps:
  76         return(b1, b2, -b1, -b2)
  77     else:
  78         if b3s - b2s - b1s > eps: #obtuse
  79             b3 = b2
  80             b2 = b2 + b1
  81         return (b1, b2, b3, -b1, -b2, -b3)
  82
  83
  84 def is_obtuse(b1, b2, tolerance=1e-13):
  85     b1s = np.sum(b1 ** 2)
  86     b2s = np.sum(b2 ** 2)
  87     b3 = b2 - b1
  88     b3s = np.sum(b3 ** 2)
  89     eps = tolerance * (b2s + b1s)
  90     return (b3s - b2s - b1s > eps)
  91
  92 def classifyLatticeSingle(b1, b2, tolerance=1e-13):
  93     """
  94     Given two basis vectors, returns 2D Bravais lattice type.
  95     Tolerance is relative.
  96     TODO doc
  97     """
  98     b1, b2 = reduceBasisSingle(b1, b2)
  99     b1s = np.sum(b1 ** 2)
 100     b2s = np.sum(b2 ** 2)
 101     b3 = b2 - b1
 102     b3s = np.sum(b3 ** 2)
 103     eps = tolerance * (b2s + b1s)
 104     # Avoid obtuse angle between b1 and b2. TODO This should be yet thoroughly tested.
 105     # TODO use is_obtuse here?
 106     if b3s - b2s - b1s > eps:
 107         b3 = b2
 108         b2 = b2 + b1
 109         # N. B. now the assumption |b3| >= |b2| is no longer valid
 110         #b3 = b2 - b1
 111         b2s = np.sum(b2 ** 2)
 112         b3s = np.sum(b3 ** 2)
 113     if abs(b2s - b1s) < eps or abs(b2s - b3s) < eps: # isoscele
 114         if abs(b3s - b1s) < eps:
 115             return LatticeType.EQUILATERAL_TRIANGULAR
 116         elif abs(b3s - 2 * b1s) < eps:
 117             return LatticeType.SQUARE
 118         else:
 119             return LatticeType.RHOMBIC
 120     elif abs(b3s - b2s - b1s) < eps:
 121         return LatticeType.RECTANGULAR
 122     else:
 123         return LatticeType.OBLIQUE
 124
 125 def range2D(maxN, mini=1, minj=0, minN = 0):
 126     """
 127     "Triangle indices"
 128     Generates pairs of non-negative integer indices (i, j) such that
 129     minN ≤ i + j ≤ maxN, i ≥ mini, j ≥ minj.
 130     TODO doc and possibly different orderings
 131     """
 132     for maxn in range(min(mini, minj, minN), floor(maxN+1)): # i + j == maxn
 133         for i in range(mini, maxn + 1):
 134             yield (i, maxn - i)
 135
 136
 137 def generateLattice(b1, b2, maxlayer=5, include_origin=False, order='leaves'):
 138     bvs = shortestBase46(b1, b2)
 139     cc = len(bvs) # "corner count"
 140
 141     if order == 'leaves':
 142         indices = np.array(list(range2D(maxlayer)))
 143         ia = indices[:,0]
 144         ib = indices[:,1]
 145         cc = len(bvs) # 4 for square/rec,
 146         leaves = list()
 147         if include_origin: leaves.append(np.array([[0,0]]))
 148         for c in range(cc):
 149             ba = bvs[c]
 150             bb = bvs[(c+1)%cc]
 151             leaves.append(ia[:,nx]*ba + ib[:,nx]*bb)
 152         return np.concatenate(leaves)
 153     else:
 154         raise ValueError('Lattice point order not implemented: ', order)
 155
 156 def generateLatticeDisk(b1, b2, r, include_origin=False, order='leaves'):
 157     b1, b2 = reduceBasisSingle(b1,b2)
 158     blen = np.linalg.norm(b1, ord=2)
 159     maxlayer = 2*r/blen # FIXME kanon na vrabce? Nestačí odmocnina ze 2?
 160     points = generateLattice(b1,b2, maxlayer=maxlayer, include_origin=include_origin, order=order)
 161     mask = (np.linalg.norm(points, axis=-1, ord=2) <= r)
 162     return points[mask]
 163
 164 def cellCornersWS(b1, b2,):
 165     """
 166     Given basis vectors, returns the corners of the Wigner-Seitz unit cell
 167     (w1, w2, -w1, w2) for rectangular and square lattice or
 168     (w1, w2, w3, -w1, -w2, -w3) otherwise
 169     """
 170     def solveWS(v1, v2):
 171         v1x = v1[0]
 172         v1y = v1[1]
 173         v2x = v2[0]
 174         v2y = v2[1]
 175         lsm = ((-v1y, v2y), (v1x, -v2x))
 176         rs = ((v1x-v2x)/2, (v1y - v2y)/2)
 177         t = np.linalg.solve(lsm, rs)
 178         return np.array(v1)/2 + t[0]*np.array((v1y, -v1x))
 179     b1, b2 = reduceBasisSingle(b1, b2)
 180     latticeType = classifyLatticeSingle(b1, b2)
 181     if latticeType is LatticeType.RECTANGULAR or latticeType is LatticeType.SQUARE:
 182         return np.array( (
 183             (+b1+b2),
 184             (+b2-b1),
 185             (-b1-b2),
 186             (-b2+b1),
 187         )) / 2
 188     else:
 189         bvs = shortestBase46(b1,b2,tolerance=0)
 190         return np.array([solveWS(bvs[i], bvs[(i+1)%6]) for i in range(6)])
 191
 192 def cutWS(points, b1, b2, scale=1., tolerance=1e-13):
 193     """
 194     From given points, return only those that are inside (or on the edge of)
 195     the Wigner-Seitz cell of a (scale*b1, scale*b2)-based lattice.
 196     """
 197     # TODO check input dimensions?
 198     bvs = shortestBase46(b1, b2)
 199     points = np.array(points)
 200     for b in bvs:
 201         mask = (np.tensordot(points, b, axes=(-1, 0)) <=  (scale * (1+tolerance) / 2) *np.linalg.norm(b, ord=2)**2 )
 202         points = points[mask]
 203     return points
 204
 205 def filledWS(b1, b2, density=10, scale=1.):
 206     """
 207     TODO doc
 208     TODO more intelligent generation, anisotropy balancing etc.
 209     """
 210     points = generateLattice(b1,b2,maxlayer=density*scale, include_origin=True)
 211     points = cutWS(points/density, np.array(b1)*scale, np.array(b2)*scale)
 212     return points
 213
 214
 215 def reciprocalBasis1(*pargs):
 216     a = reduceBasisSingle(*pargs)
 217     return np.linalg.inv(a).T
 218
 219 def reciprocalBasis2pi(*pargs):
 220     return 2*np.pi*reciprocalBasis1(*pargs)
 221
 222 # TODO fill it with "points from reciprocal space" instead
 223 def filledWS2(b1,b2, density=10, scale=1.):
 224     b1, b2 = reduceBasisSingle(b1,b2)
 225     b1r, b2r = reciprocalBasis2pi(b1,b2)
 226     b1l = np.linalg.norm(b1, ord=2)
 227     b2l = np.linalg.norm(b2, ord=2)
 228     b1rl = np.linalg.norm(b1r, ord=2)
 229     b2rl = np.linalg.norm(b2r, ord=2)
 230     # Black magick. Think later.™ Really. FIXME
 231     sicher_ratio = np.maximum(b1rl/b2rl, b2rl/b1rl) * np.maximum(b1l/b2l, b2l/b1l) # This really has to be adjusted
 232     points = generateLattice(b1r,b2r,maxlayer=density*scale*sicher_ratio, include_origin=True)
 233     points = cutWS(points*b1l/b1rl/density, b1*scale, b2*scale)
 234     return points
 235
 236
 237 def change_basis(srcbasis, destbasis, srccoords, srccoordsaxis=-1, lattice=True):
 238     srcbasis = np.array(srcbasis)
 239     destbasis = np.array(destbasis)
 240     trmatrix = np.dot(np.linalg.inv(np.transpose(destbasis)), np.transpose(srcbasis))
 241     if lattice: # if srcbasis and destbasis are two bases of the same lattice, its elements are ints
 242         otrmatrix = trmatrix
 243         trmatrix = np.round(trmatrix)
 244         if not np.all(np.isclose(trmatrix, otrmatrix)):
 245             raise ValueError("Given srcbasis and destbasis are not bases"
 246                 "of the same lattice", srcbasis, destbasis, trmatrix-otrmatrix)
 247     destcoords = np.tensordot(srccoords, trmatrix, axes=(srccoordsaxis, -1))
 248     return destcoords
 249
 250 """
 251 TODO
 252 ====
 253
 254 - DOC!!!!!
 255 - (nehoří) výhledově pořešit problém „hodně anisotropních“ mřížek (tj. kompensovat
 256 rozdílné délky základních vektorů).
 257
 258 """