Fix module path
Former-commit-id: 6b8d455aae082bde323031b7eb57268701f18c77
This commit is contained in:
Marek Nečada
parent
13556b210d
commit
cc4815b861
247
lattices2d.py
247
lattices2d.py
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import numpy as np
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from enum import Enum
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nx = None
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class LatticeType(Enum):
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"""
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All the five Bravais lattices in 2D
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"""
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OBLIQUE=1
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RECTANGULAR=2
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SQUARE=4
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RHOMBIC=5
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EQUILATERAL_TRIANGULAR=3
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RIGHT_ISOSCELES=SQUARE
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PARALLELOGRAMMIC=OBLIQUE
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CENTERED_RHOMBIC=RECTANGULAR
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RIGHT_TRIANGULAR=RECTANGULAR
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CENTERED_RECTANGULAR=RHOMBIC
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ISOSCELE_TRIANGULAR=RHOMBIC
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RIGHT_ISOSCELE_TRIANGULAR=SQUARE
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HEXAGONAL=EQUILATERAL_TRIANGULAR
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def reduceBasisSingle(b1, b2):
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"""
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Lagrange-Gauss reduction of a 2D basis.
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cf. https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch17.pdf
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inputs and outputs are (2,)-shaped numpy arrays
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The output shall satisfy |b1| <= |b2| <= |b2 - b1|
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TODO doc
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TODO perhaps have the (on-demand?) guarantee of obtuse angle between b1, b2?
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TODO possibility of returning the (in-order, no-obtuse angles) b as well?
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"""
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b1 = np.array(b1)
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b2 = np.array(b2)
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if b1.shape != (2,) or b2.shape != (2,):
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raise ValueError('Shape of b1 and b2 must be (2,)')
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B1 = np.sum(b1 * b1, axis=-1, keepdims=True)
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mu = np.sum(b1 * b2, axis=-1, keepdims=True) / B1
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b2 = b2 - np.rint(mu) * b1
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B2 = np.sum(b2 * b2, axis=-1, keepdims=True)
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while(np.any(B2 < B1)):
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b2t = b1
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b1 = b2
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b2 = b2t
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B1 = B2
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mu = np.sum(b1 * b2, axis=-1, keepdims=True) / B1
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b2 = b2 - np.rint(mu) * b1
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B2 = np.sum(b2*b2, axis=-1, keepdims=True)
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return(b1,b2)
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def shortestBase3(b1, b2):
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'''
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returns the "ordered shortest triple" of base vectors (each pair from
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the triple is a base) and there may not be obtuse angle between b1, b2
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and between b2, b3
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'''
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b1, b2 = reduceBasisSingle(b1,b2)
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if is_obtuse(b1, b2, tolerance=0):
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b3 = b2
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b2 = b2 + b1
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else:
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b3 = b2 - b1
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return (b1, b2, b3)
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def shortestBase46(b1, b2, tolerance=1e-13):
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b1, b2 = reduceBasisSingle(b1,b2)
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b1s = np.sum(b1 ** 2)
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b2s = np.sum(b2 ** 2)
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b3 = b2 - b1
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b3s = np.sum(b3 ** 2)
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eps = tolerance * (b2s + b1s)
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if abs(b3s - b2s - b1s) < eps:
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return(b1, b2, -b1, -b2)
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else:
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if b3s - b2s - b1s > eps: #obtuse
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b3 = b2
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b2 = b2 + b1
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return (b1, b2, b3, -b1, -b2, -b3)
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def is_obtuse(b1, b2, tolerance=1e-13):
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b1s = np.sum(b1 ** 2)
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b2s = np.sum(b2 ** 2)
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b3 = b2 - b1
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b3s = np.sum(b3 ** 2)
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eps = tolerance * (b2s + b1s)
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return (b3s - b2s - b1s > eps)
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def classifyLatticeSingle(b1, b2, tolerance=1e-13):
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"""
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Given two basis vectors, returns 2D Bravais lattice type.
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Tolerance is relative.
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TODO doc
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"""
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b1, b2 = reduceBasisSingle(b1, b2)
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b1s = np.sum(b1 ** 2)
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b2s = np.sum(b2 ** 2)
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b3 = b2 - b1
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b3s = np.sum(b3 ** 2)
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eps = tolerance * (b2s + b1s)
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# Avoid obtuse angle between b1 and b2. TODO This should be yet thoroughly tested.
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# TODO use is_obtuse here?
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if b3s - b2s - b1s > eps:
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b3 = b2
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b2 = b2 + b1
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# N. B. now the assumption |b3| >= |b2| is no longer valid
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#b3 = b2 - b1
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b2s = np.sum(b2 ** 2)
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b3s = np.sum(b3 ** 2)
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if abs(b2s - b1s) < eps or abs(b2s - b3s) < eps: # isoscele
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if abs(b3s - b1s) < eps:
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return LatticeType.EQUILATERAL_TRIANGULAR
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elif abs(b3s - 2 * b1s) < eps:
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return LatticeType.SQUARE
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else:
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return LatticeType.RHOMBIC
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elif abs(b3s - b2s - b1s) < eps:
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return LatticeType.RECTANGULAR
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else:
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return LatticeType.OBLIQUE
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def range2D(maxN, mini=1, minj=0, minN = 0):
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"""
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"Triangle indices"
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Generates pairs of non-negative integer indices (i, j) such that
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minN ≤ i + j ≤ maxN, i ≥ mini, j ≥ minj.
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TODO doc and possibly different orderings
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"""
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for maxn in range(min(mini, minj, minN), floor(maxN+1)): # i + j == maxn
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for i in range(mini, maxn + 1):
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yield (i, maxn - i)
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def generateLattice(b1, b2, maxlayer=5, include_origin=False, order='leaves'):
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bvs = shortestBase46(b1, b2)
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cc = len(bvs) # "corner count"
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if order == 'leaves':
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indices = np.array(list(range2D(maxlayer)))
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ia = indices[:,0]
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ib = indices[:,1]
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cc = len(bvs) # 4 for square/rec,
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leaves = list()
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if include_origin: leaves.append(np.array([[0,0]]))
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for c in range(cc):
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ba = bvs[c]
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bb = bvs[(c+1)%cc]
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leaves.append(ia[:,nx]*ba + ib[:,nx]*bb)
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return np.concatenate(leaves)
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else:
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raise ValueError('Lattice point order not implemented: ', order)
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def generateLatticeDisk(b1, b2, r, include_origin=False, order='leaves'):
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b1, b2 = reduceBasisSingle(b1,b2)
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blen = np.linalg.norm(b1, ord=2)
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maxlayer = 2*r/blen # FIXME kanon na vrabce? Nestačí odmocnina ze 2?
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points = generateLattice(b1,b2, maxlayer=maxlayer, include_origin=include_origin, order=order)
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mask = (np.linalg.norm(points, axis=-1, ord=2) <= r)
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return points[mask]
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def cellCornersWS(b1, b2,):
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"""
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Given basis vectors, returns the corners of the Wigner-Seitz unit cell
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(w1, w2, -w1, w2) for rectangular and square lattice or
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(w1, w2, w3, -w1, -w2, -w3) otherwise
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"""
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def solveWS(v1, v2):
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v1x = v1[0]
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v1y = v1[1]
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v2x = v2[0]
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v2y = v2[1]
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lsm = ((-v1y, v2y), (v1x, -v2x))
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rs = ((v1x-v2x)/2, (v1y - v2y)/2)
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t = np.linalg.solve(lsm, rs)
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return np.array(v1)/2 + t[0]*np.array((v1y, -v1x))
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b1, b2 = reduceBasisSingle(b1, b2)
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latticeType = classifyLatticeSingle(b1, b2)
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if latticeType is LatticeType.RECTANGULAR or latticeType is LatticeType.SQUARE:
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return np.array( (
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(+b1+b2),
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(+b2-b1),
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(-b1-b2),
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(-b2+b1),
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)) / 2
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else:
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bvs = shortestBase46(b1,b2,tolerance=0)
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return np.array([solveWS(bvs[i], bvs[(i+1)%6]) for i in range(6)])
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def cutWS(points, b1, b2, scale=1., tolerance=1e-13):
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"""
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From given points, return only those that are inside (or on the edge of)
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the Wigner-Seitz cell of a (scale*b1, scale*b2)-based lattice.
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"""
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# TODO check input dimensions?
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bvs = shortestBase46(b1, b2)
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points = np.array(points)
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for b in bvs:
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mask = (np.tensordot(points, b, axes=(-1, 0)) <= (scale * (1+tolerance) / 2) *np.linalg.norm(b, ord=2)**2 )
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points = points[mask]
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return points
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def filledWS(b1, b2, density=10, scale=1.):
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"""
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TODO doc
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TODO more intelligent generation, anisotropy balancing etc.
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"""
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points = generateLattice(b1,b2,maxlayer=density*scale, include_origin=True)
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points = cutWS(points/density, np.array(b1)*scale, np.array(b2)*scale)
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return points
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rot90_ = np.array([[0,1],[-1,0]])
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def reciprocalBasis(a1, a2):
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a1, a2 = reduceBasisSingle(a1,a2) # this can be replaced with the vector version of reduceBasis when it is made
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prefac = 2*np.pi/np.sum(np.tensordot(a1, rot90_, axes=[-1,0]) * a2, axis=-1)
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b1 = np.tensordot(rot90_, a2, axes=[-1,-1]) * prefac
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b2 = np.tensordot(rot90_, a1, axes=[-1,-1]) * prefac
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return (b1, b2)
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# TODO fill it with "points from reciprocal space" instead
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def filledWS2(b1,b2, density=10, scale=1.):
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b1, b2 = reduceBasisSingle(b1,b2)
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b1r, b2r = reciprocalBasis(b1,b2)
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b1l = np.linalg.norm(b1, ord=2)
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b2l = np.linalg.norm(b2, ord=2)
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b1rl = np.linalg.norm(b1r, ord=2)
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b2rl = np.linalg.norm(b2r, ord=2)
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# Black magick. Think later.™ Really. FIXME
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sicher_ratio = np.maximum(b1rl/b2rl, b2rl/b1rl) * np.maximum(b1l/b2l, b2l/b1l) # This really has to be adjusted
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points = generateLattice(b1r,b2r,maxlayer=density*scale*sicher_ratio, include_origin=True)
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points = cutWS(points*b1l/b1rl/density, b1*scale, b2*scale)
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return points
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"""
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TODO
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====
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- DOC!!!!!
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- (nehoří) výhledově pořešit problém „hodně anisotropních“ mřížek (tj. kompensovat
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rozdílné délky základních vektorů).
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"""
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@ -152,6 +152,14 @@ def generateLattice(b1, b2, maxlayer=5, include_origin=False, order='leaves'):
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else:
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raise ValueError('Lattice point order not implemented: ', order)
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def generateLatticeDisk(b1, b2, r, include_origin=False, order='leaves'):
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b1, b2 = reduceBasisSingle(b1,b2)
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blen = np.linalg.norm(b1, ord=2)
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maxlayer = 2*r/blen # FIXME kanon na vrabce? Nestačí odmocnina ze 2?
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points = generateLattice(b1,b2, maxlayer=maxlayer, include_origin=include_origin, order=order)
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mask = (np.linalg.norm(points, axis=-1, ord=2) <= r)
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return points[mask]
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def cellCornersWS(b1, b2,):
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"""
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Given basis vectors, returns the corners of the Wigner-Seitz unit cell
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