Hyperbolic Geometry Utility Functions

Imports

from typing import List

import numpy as np
import matplotlib.pyplot as plt

from pyzeta.geometry.sl2r import SL2R
from pyzeta.geometry.visuals import plotFundDom, hypPlaneWave, SLtoSU
from pyzeta.geometry.helpers import HtoD
from pyzeta.core.dynamics.function_systems.implementations import (
    FunnelTorus,
    HyperbolicCylinder,
)
from pyzeta.framework.initialization.initialization_handler import (
    PyZetaInitializationHandler,
)

PyZetaInitializationHandler.initPyZetaServices()

Define the Examples

def funnelTorusExample(model: str = "H") -> None:
    "Illustrate fundamental domain plotting with funneled torus."
    torus = FunnelTorus(2.0, 2.0, np.pi / 2.0)
    z0 = 1j

    generators = [SL2R(mat) for mat in torus._phi[:2]]
    if model == "D":
        for i, gen in enumerate(generators):
            generators[i] = SLtoSU(gen)  # type: ignore
        z0 = HtoD(z0)
    plotFundDom(*generators, z0=z0)
    plt.show()


def hyperbolicCylinderExample(model: str = "H") -> None:
    "Illustrate fundamental domain plotting with funneled torus."
    torus = HyperbolicCylinder(3.0, rotate=True)
    z0 = 1j

    generators = [SL2R(mat) for mat in torus._phi[:1]]
    if model == "D":
        for i, gen in enumerate(generators):
            generators[i] = SLtoSU(gen)  # type: ignore
        z0 = HtoD(z0)
    plotFundDom(*generators, z0=z0)
    plt.show()

def planeWavesExample1(model: str = "H") -> None:
    "Simple example of hyperbolic plane waves on the upper halfplane."
    # hyperbolic plane waves example1
    realArr = np.linspace(-2.0, 2.0, 1000)
    imagArr = np.linspace(0.0, 4.0, 1000)
    xi: List[complex] = [
        0.0 + 0.0j,
    ]
    k = [
        5.0,
    ]

    wave = hypPlaneWave(realArr, imagArr, xi, k, model="H")
    wave[wave == np.infty] = -1.0
    plt.imshow(
        np.abs(wave),
        cmap="Reds",
        interpolation="nearest",
        origin="lower",
        extent=[-2, 2, 0, 4.0],
    )
    plt.show()

def planeWavesExample2(model: str = "H") -> None:
    "More complex example of hyperbolic plane waves on the Poincare disc."
    realArr = np.linspace(-1.0, 1.0, 8000)
    imagArr = np.linspace(-1.0, 1.0, 8000)
    xi = [1.0, -1.0]
    k = [2.0, 3.0]

    wave = hypPlaneWave(realArr, imagArr, xi, k, model="D")
    wave[wave == np.nan] = 0.0
    plt.imshow(
        np.abs(wave),
        cmap="Reds",
        interpolation="nearest",
        origin="lower",
        extent=[-1, 1, -1, 1],
    )
    plt.show()

Run the Examples

funnelTorusExample(model="H")
hyperbolicCylinderExample(model="H")
funnelTorusExample(model="D")
hyperbolicCylinderExample(model="D")

planeWavesExample1()

planeWavesExample2()