""" Example flute surfaces. """
from __future__ import annotations
import bigger
from bigger.types import FlatTriangle
from bigger.triangulation import Triangle
from .utils import integers, extract_curve_and_test
Edge = int
[docs]def flute() -> bigger.MCG[Edge]:
"""The infinitely punctured sphere, with punctures that accumulate in one direction.
With mapping classes:
- a_n which twists about the curve parallel to edges n and n+1
- b_n which twists about the curve which separates punctures n and n+1
- a{expr(n)} which twists about all a_n curves when expr(n) is True
- b{expr(n)} which twists about all b_n curves when expr(n) is True
Shortcuts:
- a[start:stop:step] = a{n in range(start, stop, step)}
- a == a[:]
"""
# #----2----#----5----#----8----#---
# /| /| /| /|
# -1 | / | / | / |
# # 0 1 3 4 6 7 9 ...
# -1 | / | / | / |
# \|/ |/ |/ |
# #----2----#----5----#----8----#---
T = bigger.Triangulation.from_pos(
lambda: integers(-1),
lambda edge: (0, True, -1, False, -1, True, 0, True)
if edge == -1
else (-1, False, -1, True, 1, True, 2, False)
if edge == 0
else [
(edge - 2, False, edge - 1, True, edge + 1, True, edge + 2, False),
(edge + 1, False, edge - 1, False, edge + 1, True, edge + 2, True),
(edge + 1, True, edge - 1, False, edge - 2, False, edge - 1, True),
][edge % 3],
)
def generator(name: str) -> bigger.Encoding[Edge]:
curve, test = extract_curve_and_test("ab", name)
if curve == "a":
return T(lambda n: 1 if n >= 0 and n % 3 != 0 and test(n // 3) else 0).twist()
if curve == "b":
def weight(n: Edge) -> int:
N, r = divmod(n, 3)
if n == -1:
return 1 if test(N + 1) else 0
elif n == 0:
return 2 if test(N) else 0
elif n == 1:
if test(N):
return 2
return 1 if test(N + 1) else 0
elif n == 2:
return 1 if test(N + 1) else 0
if r == 0:
return 2 if test(N) or test(N - 1) else 0
elif r == 1:
if test(N) or (test(N - 1) and test(N + 1)):
return 2
return 1 if test(N - 1) or test(N + 1) else 0
else: # r == 2:
if test(N):
return 0
if test(N - 1) and test(N + 1):
return 2
return 1 if test(N - 1) or test(N + 1) else 0
return T(weight).twist()
raise ValueError(f"Unknown mapping class {name}")
def layout(triangle: Triangle[Edge]) -> FlatTriangle:
if triangle[0].edge == -1:
return (0.0, 1.0), (-1.0, 0.5), (0.0, 0.0)
elif triangle[0].edge % 3 == 0:
n = triangle[0].edge // 3
return (n, 1.0), (n, 0.0), (n + 1.0, 1.0)
else: # triangle[0] % 3 == 1:
n = triangle[0].edge // 3
return (n + 1.0, 1.0), (n, 0.0), (n + 1.0, 0.0)
return bigger.MCG(T, generator, layout)
[docs]def biflute() -> bigger.MCG[Edge]:
"""The infinitely punctured sphere, with punctures that accumulate in two directions.
With mapping classes:
- a_n which twists about the curve parallel to edges n and n+1
- b_n which twists about the curve which separates punctures n and n+1
- a{expr(n)} which twists about all a_n curves when expr(n) is True
- b{expr(n)} which twists about all b_n curves when expr(n) is True
- s which shifts the surface down
- r which rotates the surface fixing the curve a_0
Shortcuts:
- a[start:stop:step] = a{n in range(start, stop, step)}
- a == a[:]
Note: Since b_n and b_{n+1} intersect, any b expression cannot be true for consecutive values.
"""
# ---#----2----#----5----#----8----#---
# | /| /| /|
# | / | / | / |
# ... 0 1 3 4 6 7 9 ...
# | / | / | / |
# |/ |/ |/ |
# ---#----2----#----5----#----8----#---
T = bigger.Triangulation.from_pos(
integers,
lambda edge: [
(edge - 2, False, edge - 1, True, edge + 1, True, edge + 2, False),
(edge + 1, False, edge - 1, False, edge + 1, True, edge + 2, True),
(edge + 1, True, edge - 1, False, edge - 2, False, edge - 1, True),
][edge % 3],
)
def generator(name: str) -> bigger.Encoding[Edge]:
if name in ("s", "shift"):
return T.isometry(T, lambda edge: edge + 3, lambda edge: edge - 3)
if name in ("r", "rotate"):
return T.isometry(T, lambda edge: [3, 2, 4][edge % 3] - edge, lambda edge: [3, 2, 4][edge % 3] - edge)
curve, test = extract_curve_and_test("ab", name)
if curve == "a":
return T(lambda n: 1 if n % 3 != 0 and test(n // 3) else 0).twist()
if curve == "b":
def weight(n: Edge) -> int:
N, r = divmod(n, 3)
if r == 0:
return 2 if test(N) or test(N - 1) else 0
elif r == 1:
if test(N) or (test(N - 1) and test(N + 1)):
return 2
return 1 if test(N - 1) or test(N + 1) else 0
else: # r == 2:
if test(N):
return 0
if test(N - 1) and test(N + 1):
return 2
return 1 if test(N - 1) or test(N + 1) else 0
return T(weight).twist()
raise ValueError(f"Unknown mapping class {name}")
def layout(triangle: Triangle[Edge]) -> FlatTriangle:
n, k = divmod(triangle[0].edge, 3)
if k == 0:
return (n, 1.0), (n, 0.0), (n + 1.0, 1.0)
else: # k == 1.
return (n + 1.0, 1.0), (n, 0.0), (n + 1.0, 0.0)
return bigger.MCG(T, generator, layout)
