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_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_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 test_apply_base_map_unit(self):
"""Test that the application of a sequence of base_maps with unit product does not change the curve."""
rot_90 = BaseMap([(1,True), (0,False)]) # поворот на 90 градусов
rot_90_t = BaseMap.parse('jI~')
bms_list = [
[rot_90] * 3,
[rot_90_t] * 3,
[
BaseMap([(1,False), (0,False)], time_rev=True),
BaseMap([(0,True), (1,False)]),
],
[
BaseMap([(0,True),(2,False),(1,True)], time_rev=True),
BaseMap([(1,False),(0,True),(2,True)], time_rev=True),
],
]
for bms in bms_list:
# will make bms[0] * bms[1] * ... * bms[-1] * last_map = id <=> last_map = bms[-1]^{-1} * ... * bms[0]^{-1}
last_map = BaseMap.id_map(dim=bms[0].dim)
for bm in bms:
last_map = bm**(-1) * last_map
for curve in self.curves:
if curve.dim != bms[0].dim:
continue
orig = curve
current = curve
for bm in reversed(bms + [last_map]):
current = bm * current
self.assertEqual(orig.proto, current.proto)
self.assertEqual(orig.specs, current.specs)
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_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_scepin_bauman_curve():
proto = (
(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([(1, True), (0, False)]), # rot(90)
BaseMap.parse('(x,y)->(y,x)'),
BaseMap.parse('(x,y)->(x,1-y)'),
BaseMap.parse('(x,y)->(1-y,1-x)'),
BaseMap([(1, False), (0, True)]), # rot(-90)
BaseMap.id_map(dim=2),
BaseMap([(1, True), (0, False)]), # rot(90)
BaseMap([(1, False), (0, False)]), # (x,y)->(y,x)
]
return Curve(dim=2, div=3, patterns=[(proto, base_maps)])
def get_rev_curve():
"""Curve with time reversal at some middle cube."""
return Curve(
dim=2,
div=2,
patterns=[
(
[(0, 0), (0, 1), (1, 1), (1, 0)],
[
BaseMap([(1, False), (0, False)]), # (x,y)->(y,x)
BaseMap([(0, True), (1, False)],
time_rev=True), # (x,y)->(1-x,y), t->1-t
BaseMap.id_map(2), # (x,y)->(x,y)
BaseMap([(1, True), (0, True)]), # (x,y)->(1-y,1-x)
],
),
],
)
def get_peano_curve():
"""Example of fractal curve due to G.Peano."""
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), (1, 2), (1, 1), (1, 0), (2, 0), (2, 1),
(2, 2)]
base_maps = [
id_map,
x_map,
id_map,
y_map,
xy_map,
y_map,
id_map,
x_map,
id_map,
]
return Curve(dim=2, div=3, patterns=[(proto, base_maps)])
def test_inv(self):
for bm in self.base_maps:
self.assertEqual(bm * ~bm, BaseMap.id_map(dim=bm.dim))
self.assertEqual(~bm * bm, BaseMap.id_map(dim=bm.dim))
def get_peano5_curve():
"""5-div analog of original Peano curve."""
id_map = BaseMap.id_map(2)
x_map = BaseMap.parse('Ij')
y_map = BaseMap.parse('iJ')
xy_map = BaseMap.parse('IJ')
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)])