def get_peano5_curve():
id_map = BaseMap.id_map(2)
x_map = BaseMap([(0, True), (1, False)]) # (x,y)->(1-x,y)
y_map = BaseMap([(0, False), (1, True)]) # (x,y)->(x,1-y)
xy_map = BaseMap([(0, True), (1, True)]) # (x,y)->(1-x,1-y)
proto = [
(0, 0),
(0, 1),
(0, 2),
(0, 3),
(0, 4),
(1, 4),
(1, 3),
(1, 2),
(1, 1),
(1, 0),
(2, 0),
(2, 1),
(2, 2),
(2, 3),
(2, 4),
(3, 4),
(3, 3),
(3, 2),
(3, 1),
(3, 0),
(4, 0),
(4, 1),
(4, 2),
(4, 3),
(4, 4),
]
base_maps = [
id_map,
x_map,
id_map,
x_map,
id_map,
y_map,
xy_map,
y_map,
xy_map,
y_map,
id_map,
x_map,
id_map,
x_map,
id_map,
y_map,
xy_map,
y_map,
xy_map,
y_map,
id_map,
x_map,
id_map,
x_map,
id_map,
]
return Curve(dim=2, div=5, patterns=[(proto, base_maps)])
def get_scepin_bauman_curve():
"""
Minimal 3*3 Peano Curve by E.V. Shchepin and K.E. Bauman.
Proceedings of the Steklov Institute of Mathematics, 2008, Vol. 263, pp. 236--256.
"""
proto = ( # as in peano curve
(0, 0),
(0, 1),
(0, 2),
(1, 2),
(1, 1),
(1, 0),
(2, 0),
(2, 1),
(2, 2),
)
base_maps = [
BaseMap.id_map(dim=2),
BaseMap.parse('jI'),
BaseMap.parse('ji'),
BaseMap.parse('iJ'),
BaseMap.parse('JI'),
BaseMap.parse('Ji'),
BaseMap.id_map(dim=2),
BaseMap.parse('jI'),
BaseMap.parse('ji'),
]
return Curve(dim=2, div=3, patterns=[(proto, base_maps)])
def get_beta_Omega_Curve():
chain_code_list = ['jiJ', 'jiJ']
bases_list = [['1iJ', '1Ji', '1ji~', '1IJ~'],
['1iJ', '1Ji', '1ji~', '0jI']]
return Curve(dim=2,
div=2,
patterns=get_patterns(chain_code_list, bases_list))
def get_17_curve():
"""
3D bifractal facet-gated curve with l2-ratio <17
"""
p0 = ('jikIJiK', '1KIJ~,0KIj,1kji,0Jki~,1JkI,0kIJ,1KJi,1JiK')
p1 = ('jkJijKJ', '1KIJ~,0ijK~,0Ikj~,0KJI~,0kiJ~,0Ijk~,0IjK,1iKJ')
return Curve.parse([p0, p1])
def get_rev_curves():
chain = 'jiJ'
bases_list = [
'ji,Ij~,ij,JI', # time rev at the middle
'Ji~,ij,ij,JI', # time rev at first cube
'ji,ij,ij,jI~', # time rev at last cube
]
return [Curve.parse([(chain, b)]) for b in bases_list]
def get_rev_curves():
chain = 'jiJ'
bases_list = [
['ji','Ij~','ij','JI'], # time rev at the middle
['Ji~','ij','ij','JI'], # time rev at first cube
['ji','ij','ij','jI~'], # time rev at last cube
]
return [Curve.parse([(chain, b)]) for b in bases_list]
def get_hilbert_curve():
"""
Example of fractal curve due to D.Hilbert.
This curve has minimal known L_1 ratio (9).
"""
pattern = ('jiJ', 'ji,ij,ij,JI')
return Curve.parse([pattern])
def get_scepin_bauman_curve():
"""
Minimal 3*3 Peano Curve by E.V. Shchepin and K.E. Bauman.
Proceedings of the Steklov Institute of Mathematics, 2008, Vol. 263, pp. 236--256.
"""
pattern = ('jjiJJijj', 'ij,jI,ji,iJ,JI,Ji,ij,jI,ji')
return Curve.parse([pattern])
def get_hilbert_curve():
"""
Example of fractal curve due to D.Hilbert.
This curve has minimal known L_1 ratio (9).
"""
pattern = ('jiJ', ['ji', 'ij', 'ij', 'JI']) # chain code + bases
return Curve.parse([pattern])
def get_tokarev_curve():
"""
3D monofractal curve defined by Tokarev.
Definition is taken from Haverkort's inventory,
Curve A26.0000 0000.0000 0000 (page 9, Fig.5(b))
"""
p0 = ('jkJijKJ', ['jki', 'kij', 'kij', 'iJK', 'iJK', 'KIj', 'KIj', 'JkI'])
return Curve.parse([p0])
def get_luna_curve():
chain_code = ['kjKikJK', 'kiKjIki']
bases = [['1ijk', '0KJi', '1KiJ', '1jKI', '1jik', '1IKj', '0kJI', '1kJI'],
['1jik', '0JKi', '1iKJ', '0KiJ', '1KjI', '1JIk', '0ikj', '1ikj']]
return Curve(
dim=3,
div=2,
patterns=get_patterns(chain_code, bases),
)
def get_neptunus_curve():
chain_code = ['kjKikJK', 'kiKjIki']
bases = [['0kji', '1kji', '1KiJ', '1jKI', '1ikj', '1KJi', '0kJI', '1jKI'],
['0jki', '1jki', '1iKJ', '0KiJ', '1JiK', '1IKj', '0ikj', '1ijk']]
return Curve(
dim=3,
div=2,
patterns=get_patterns(chain_code, bases),
)
def get_morton_curve():
chain_code = 'i', 'Ij', 'i'
bases = ['ij'] * 4
return Curve(
dim=2,
div=2,
patterns=[(chain2proto(chain_code), [basis2base_map(b)
for b in bases])],
)
def get_discontinuous_curve():
proto = [(1, 1), (0, 1), (0, 0), (1, 0), (2, 0), (2, 1), (2, 2), (1, 2),
(0, 2)]
return Curve(
dim=2,
div=2,
patterns=[
(proto, [BaseMap.id_map(2)] * 9),
],
)
def get_R_curve():
chain_code = 'jjiiJIJi'
bases = ['ji', 'ji', 'ij', 'ij', 'ij', 'IJ', 'JI', 'JI', 'ij']
return Curve(
dim=2,
div=3,
patterns=[
(chain2proto(chain_code), [basis2base_map(b) for b in bases]),
],
)
def get_hilbert_curve():
"""Example of fractal curve due to D.Hilbert."""
proto = [(0, 0), (0, 1), (1, 1), (1, 0)]
base_maps = [
BaseMap.parse('(x,y)->(y,x)'),
BaseMap.id_map(2),
BaseMap.id_map(2),
BaseMap.parse('(x,y)->(1-y,1-x)'),
]
return Curve(dim=2, div=2, patterns=[(proto, base_maps)])
def get_serpentine_curve():
chain_code = 'jjiJJijj'
bases = ['ij', 'Ji', 'ji', 'iJ', 'JI', 'iJ', 'ji', 'Ji', 'ij']
return Curve(
dim=2,
div=3,
patterns=[
(chain2proto(chain_code), [basis2base_map(b) for b in bases]),
],
)
def get_luna_curve():
"""
3D bifractal Luna curve
Inventory ref, p.16
gates: (0,0,0)->(1,0,0), (0,0,0)->(1,1,1)
"""
p0 = ('jkJijKJ', '1kji,0jIK,1JIk,1iKJ,1ijk,1jKI,0JiK,1KiJ')
p1 = ('jkJiKjk', '1kji,0jIK,1JIk,1iKJ,0ijk,1KjI,0kij,1jik')
return Curve.parse([p0, p1])
def get_ARW_Curve():
"""
AR^2W^2 tetrafractal curve.
See reference to Haverkort & Walderveen in get_beta_omega_curve doc
"""
r_pattern = ('i(Ij)i', ['3ij', '1Ji~', '2jI', '1iJ']) # pnum=0
f_pattern = ('jiJ', ['3ji', '2Ij~', '1ij', '1JI']) # pnum=1
p_pattern = ('jiJ', ['0ji', '1jI', '0Ji', '1JI']) # pnum=2
g_pattern = ('jiJ', ['0ij', '2jI', '0Ji', '3jI~']) # pnum=3
return Curve.parse([r_pattern, f_pattern, p_pattern, g_pattern])
def get_tokarev_curve():
"""3-D curve."""
chain_code = 'kjKikJK'
bases = ['jki', 'kij', 'kij', 'iJK', 'iJK', 'KIj', 'KIj', 'JkI']
return Curve(
dim=3,
div=2,
patterns=[
(chain2proto(chain_code), [basis2base_map(b) for b in bases]),
],
)
def get_hilbert_bicurve():
"""Fake bifractal curve, actually it is Hilber curve."""
curve = get_hilbert_curve()
p1 = (curve.proto, [
Spec(sp.base_map, pnum=cnt % 2) for cnt, sp in enumerate(curve.specs)
])
p2 = (curve.proto, [
Spec(sp.base_map, pnum=(cnt + 1) % 2)
for cnt, sp in enumerate(curve.specs)
])
return Curve(dim=2, div=2, patterns=[p1, p2])
def get_ARW_Curve():
"""
AR^2W^2 tetrafractal curve.
See reference to Haverkort & Walderveen in get_beta_omega_curve doc
"""
r_pattern = ('i(Ij)i', '3ij,1Ji~,2jI,1iJ') # pnum=0
f_pattern = ('jiJ', '3ji,2Ij~,1ij,1JI') # pnum=1
p_pattern = ('jiJ', '0ji,1jI,0Ji,1JI') # pnum=2
g_pattern = ('jiJ', '0ij,2jI,0Ji,3jI~') # pnum=3
return Curve.parse([r_pattern, f_pattern, p_pattern, g_pattern])
def get_haverkort_curve_2():
"""3-D curve with time reversal."""
chain_code = 'kjKikJK'
bases = ['KIJ~', 'KJI~', 'KjI', 'Jki~', 'jki', 'kjI~', 'kJI', 'iKJ']
return Curve(
dim=3,
div=2,
patterns=[
(chain2proto(chain_code), [basis2base_map(b) for b in bases]),
],
)
def get_rev3_curve():
"""Curve with time reversal at last cube."""
proto = [(0, 0), (0, 1), (1, 1), (1, 0)]
base_maps = [
BaseMap([(1, False), (0, False)]), # (x,y)->(y,x)
BaseMap.id_map(2), # (x,y)->(x,y)
BaseMap.id_map(2), # (x,y)->(x,y)
BaseMap([(1, True), (0, False)],
time_rev=True), # (x,y)->(1-y,x), t->1-t
]
return Curve(dim=2, div=2, patterns=[(proto, base_maps)])
def get_haverkort_curve_1():
"""3-D curve with time reversal."""
chain_code = 'kjKikJK'
bases = ['kji', 'jik', 'kIj~', 'iKJ', 'IKJ~', 'KIj', 'Kij~', 'Jki~']
return Curve(
dim=3,
div=2,
patterns=[
(chain2proto(chain_code), [basis2base_map(b) for b in bases]),
],
)
def get_luna_curve():
"""
3D bifractal Luna curve
Inventory ref, p.16
gates: (0,0,0)->(1,0,0), (0,0,0)->(1,1,1)
"""
p0 = ('jkJijKJ',
['1kji', '0jIK', '1JIk', '1iKJ', '1ijk', '1jKI', '0JiK', '1KiJ'])
p1 = ('jkJiKjk',
['1kji', '0jIK', '1JIk', '1iKJ', '0ijk', '1KjI', '0kij', '1jik'])
return Curve.parse([p0, p1])
def get_neptunus_curve():
"""
3D bifractal Neptunus curve, best in linf (9.45)
"An inventory of three-dimensional Hilbert space-filling curves", Herman Haverkort
https://arxiv.org/abs/1109.2323
p.16
Properties:
ratio: linf=9.45, best in class of poly-Hilbert curves
gates: (0,0,0)->(1,0,0), (0,0,0)->(1,1,1)
"""
p0 = ('jkJijKJ', '1ijk,0jIK,1kJI,1JiK,1ijk,1jKI,1KJi,0JIk')
p1 = ('jkJiKjk', '1ijk,0jIK,1kJI,1JiK,0ijk,1KjI,1jki,0kIJ')
return Curve.parse([p0, p1])
def get_haverkort_curve_a26():
"""
3D Haverkort A26 curve, best monofractal in linf.
Monofractal curve with time reversal.
"An inventory of three-dimensional Hilbert space-filling curves", Herman Haverkort
https://arxiv.org/abs/1109.2323
Curve A26.0010 1011.1011 0011, see p.10, p.15, p.18
Properties:
ratio: linf=12.4
gate: (0,0,0)->(1,0,0)
"""
pattern = ('jkJijKJ', 'Jki~,Kij~,kij,IKJ~,iKJ,kIj~,KIj,JIk')
return Curve.parse([pattern])
def get_haverkort_curve_A26():
"""3-D curve with time reversal."""
proto = [(0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 1, 0), (1, 1, 0), (1, 1, 1),
(1, 0, 1), (1, 0, 0)]
base_maps = [
BaseMap([(2, False), (1, False), (0, False)]),
BaseMap([(2, False), (0, False), (1, False)]),
BaseMap([(2, False), (0, True), (1, False)], time_rev=True),
BaseMap([(0, False), (2, True), (1, True)]),
BaseMap([(0, True), (2, True), (1, True)], time_rev=True),
BaseMap([(2, True), (0, True), (1, False)]),
BaseMap([(2, True), (0, False), (1, False)], time_rev=True),
BaseMap([(1, True), (2, False), (0, False)], time_rev=True),
]
return Curve(dim=3, div=2, patterns=[(proto, base_maps)])
def get_haverkort_curve_f():
"""
3D Haverkort F curve, best known monofractal in l2
Monofractal curve with time reversal,
"An inventory of three-dimensional Hilbert space-filling curves", Herman Haverkort
https://arxiv.org/abs/1109.2323
Curve F, see p.13, p.15, p.18
Properties:
ratio: l1=89.8, l2=18.6 (best known!)
gate: (0,1/3,1/3)->(2/3,1/3,0)
"""
pattern = ('jkJijKJ', 'iKJ,jIK,jIk~,JkI,Jki~,jik,jiK~,kiJ~')
return Curve.parse([pattern])