""" Example lochness 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 loch_ness_monster() -> bigger.MCG[Edge]:
"""The infinite-genus, one-ended surface.
With mapping classes:
- a which twists about the longitudes of the monster
- b which twists about the meridians of the monster
- c which twists about the curves linking the nth and n+1st handles
- s which shifts the surface down"""
# #---n+2---#
# | /|
# | / |
# n n+1 n+6
# | / |
# |/ |
# #---n+3---#
# | /|
# | / |
# n+4 n+5 n+4
# | / |
# |/ |
# #---n+2---#
def link(edge: Edge) -> tuple[Edge, bool, Edge, bool, Edge, bool, Edge, bool]:
n, k = divmod(edge, 6)
return {
0: (6 * n - 6 + 1, False, 6 * n - 6 + 3, True, 6 * n + 1, True, 6 * n + 2, False),
1: (6 * n + 2, False, 6 * n, False, 6 * n + 3, True, 6 * n + 6, True),
2: (6 * n + 4, True, 6 * n + 5, False, 6 * n, False, 6 * n + 1, True),
3: (6 * n + 6, True, 6 * n + 1, False, 6 * n + 4, False, 6 * n + 5, True),
4: (6 * n + 5, False, 6 * n + 2, True, 6 * n + 5, True, 6 * n + 3, False),
5: (6 * n + 3, False, 6 * n + 4, False, 6 * n + 2, True, 6 * n + 4, True),
}[k]
T = bigger.Triangulation[Edge].from_pos(integers, link)
shift = T.isometry(T, lambda edge: edge + 6, lambda edge: edge - 6)
def generator(name: str) -> bigger.Encoding[Edge]:
if name in ("s", "shift"):
return shift
curve, test = extract_curve_and_test("abc", name)
if curve == "a":
return T(lambda edge: 1 if edge % 6 in {1, 2, 3, 5} and test(edge // 6) else 0).twist()
if curve == "b":
return T(lambda edge: 1 if edge % 6 in {4, 5} and test(edge // 6) else 0).twist()
if curve == "c":
def c(edge: Edge) -> int:
n, k = divmod(edge, 6)
if test(n) and test(n + 1):
return {0: 2, 1: 4, 2: 2, 3: 2, 4: 2, 5: 2}[k]
elif test(n):
return {0: 2, 1: 2, 2: 0, 3: 2, 4: 1, 5: 1}[k]
elif test(n + 1):
return {0: 0, 1: 2, 2: 2, 3: 0, 4: 1, 5: 1}[k]
else:
return 0
return T(c).twist()
raise ValueError(f"Unknown mapping class {name}")
def layout(triangle: Triangle[Edge]) -> FlatTriangle:
n, k = divmod(triangle[0].edge, 6)
return {
0: ((n, 2.0), (n, 1.0), (n + 1.0, 2.0)),
1: ((n + 1.0, 2.0), (n, 1.0), (n + 1.0, 1.0)),
3: ((n + 1.0, 1.0), (n, 1.0), (n + 0.1, 0.0)),
2: ((n + 0.1, 0.0), (n + 1.0 - 0.1, 0.0), (n + 1.0, 1.0)),
}[k]
return bigger.MCG(T, generator, layout)
