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']) # chain code + bases
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')
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_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_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_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_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])
def get_beta_omega_curve():
"""
Beta-Omega bifractal 2d curve, best polyfractal in l2
See J.-M. Wierum, ``Definition of a New Circular Space-Filling Curve'',
Technical report PC^2, 2002
Realization from Haverkort & Walderveen,
``Locality and bounding-box quality of two-dimensional space-filling curves'',
Comp.Geom and Appl, v.43 (2010)
ratio: l2=5, l1=9, linf=5
gates: omega: (0,1/3)->(1,1/3), beta: (0,1/3)->(2/3,0)
"""
omega_pattern = ('jiJ', ['1iJ', '1jI', '1ji~', '1IJ~'])
beta_pattern = ('jiJ', ['1iJ', '1jI', '1ji~', '0Ji'])
return Curve.parse([omega_pattern, beta_pattern])
def get_coil_curve():
"""Coil 2D 3-div curve, see Haverkort & Walderveen."""
pattern = ('jjiJJijj',
['ji', 'jI', 'ji', 'Ji', 'JI', 'Ji', 'ji', 'jI', 'ji'])
return Curve.parse([pattern])
def get_peano5_curve():
"""5-div analog of original Peano curve."""
chain_code = 'jjjjiJJJJijjjjiJJJJijjjj'
base_maps = 'ij,Ij,ij,Ij,ij, iJ,IJ,iJ,IJ,iJ, ij,Ij,ij,Ij,ij, iJ,IJ,iJ,IJ,iJ, ij,Ij,ij,Ij,ij'
pattern = (chain_code, base_maps)
return Curve.parse([pattern])
def get_meurthe_curve():
"""Meurthe curve, equivalent to Schepin-Bauman curve."""
pattern = ('jjiJJijj', 'ji,jI,ij,Ji,JI,iJ,ji,jI,ij')
return Curve.parse([pattern])
def get_coil_curve():
"""Coil 2D 3-div curve, see Haverkort & Walderveen."""
pattern = ('jjiJJijj', 'ji,jI,ji,Ji,JI,Ji,ji,jI,ji')
return Curve.parse([pattern])
def get_peano_curve():
"""First example of space-filling curve due to G.Peano."""
chain = 'jji' + 'JJi' + 'jj'
bases = ['ij', 'Ij', 'ij', 'iJ', 'IJ', 'iJ', 'ij', 'Ij', 'ij']
pattern = (chain, bases)
return Curve.parse([pattern])
def get_serpentine_curve():
"""Serpentine 2D 3-div curve, see Haverkort & Walderveen."""
pattern = ('jjiJJijj', 'ij,jI,ji,iJ,JI,iJ,ji,jI,ij')
return Curve.parse([pattern])
def get_ye_curve():
"""YE-curve: 5*5 monofractal with l2-ratio 5 43/73."""
chain = 'jiJijjIIjjiJijiiJIJiJIJi'
bases = '0Ji~,0ij,0ij,0jI~,0Ji~,0Ji~,0Ji~,0IJ,0IJ,0Ji~,0ij,0ij,0jI~,0Ji~,0ij,0ij,0ij,0IJ,0jI~,0jI~,0ij,0IJ,0jI~,0jI~,0ij'
return Curve.parse([(chain, bases)])
def get_r_curve():
"""R-curve, 2D 3-div, see Haverkort & Walderveen."""
pattern = ('jjiiJIJi',
['ji', 'ji', 'ij', 'ij', 'ij', 'IJ', 'JI', 'JI', 'ij'])
return Curve.parse([pattern])
def get_serpentine_curve():
"""Serpentine 2D 3-div curve, see Haverkort & Walderveen."""
pattern = ('jjiJJijj',
['ij', 'jI', 'ji', 'iJ', 'JI', 'iJ', 'ji', 'jI', 'ij'])
return Curve.parse([pattern])
def get_peano_curve():
"""First example of space-filling curve due to G.Peano."""
pattern = ('jjiJJijj', 'ij,Ij,ij,iJ,IJ,iJ,ij,Ij,ij')
return Curve.parse([pattern])
def get_meurthe_curve():
"""Meurthe curve, equivalent to Schepin-Bauman curve."""
pattern = ('jjiJJijj',
['ji', 'jI', 'ij', 'Ji', 'JI', 'iJ', 'ji', 'jI', 'ij'])
return Curve.parse([pattern])
def get_r_curve():
"""R-curve, 2D 3-div, see Haverkort & Walderveen."""
pattern = ('jjiiJIJi', 'ji,ji,ij,ij,ij,IJ,JI,JI,ij')
return Curve.parse([pattern])