"""
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
import numpy as np
from numpy.typing import ArrayLike
from .base import CoordinatesBase
[docs]
class BisphericalCoordinates(CoordinatesBase):
"""3-dimensional bispherical coordinates"""
dim = 3
axes = ["σ", "τ", "φ"]
_axes_alt = {"σ": ["sigma"], "τ": ["tau"], "φ": ["phi"]}
coordinate_limits = [(0, np.pi), (-np.inf, np.inf), (0, 2 * np.pi)]
def __init__(self, scale_parameter: float = 1):
super().__init__()
if scale_parameter <= 0:
raise ValueError("Scale parameter must be positive")
self.scale_parameter = scale_parameter
def __repr__(self) -> str:
"""return instance as string"""
return f"{self.__class__.__name__}(scale_parameter={self.scale_parameter})"
def __eq__(self, other):
return (
self.__class__ is other.__class__
and self.scale_parameter == other.scale_parameter
)
def _pos_to_cart(self, points: np.ndarray) -> np.ndarray:
σ, τ, φ = points[..., 0], points[..., 1], points[..., 2]
denom = np.cosh(τ) - np.cos(σ)
x = self.scale_parameter * np.sin(σ) / denom * np.cos(φ)
y = self.scale_parameter * np.sin(σ) / denom * np.sin(φ)
z = self.scale_parameter * np.sinh(τ) / denom
return np.stack((x, y, z), axis=-1)
def _pos_from_cart(self, points: np.ndarray) -> np.ndarray:
x, y, z = points[..., 0], points[..., 1], points[..., 2]
d = np.hypot(x, y) # denotes the distance from the z-axis
a = self.scale_parameter
h2 = z**2 + d**2
denom = a**2 - h2 + np.sqrt((a**2 - h2) ** 2 + 4 * a**2 * d**2)
σ = np.pi - 2 * np.arctan2(2 * a * d, denom)
τ = 0.5 * np.log(((z + a) ** 2 + d**2) / ((z - a) ** 2 + d**2))
φ = np.arctan2(y, x)
return np.stack((σ, τ, φ), axis=-1)
def _mapping_jacobian(self, points: np.ndarray) -> np.ndarray:
σ, τ, φ = points[..., 0], points[..., 1], points[..., 2]
sinσ = np.sin(σ)
cosσ = np.cos(σ)
sinhτ = np.sinh(τ)
coshτ = np.cosh(τ)
sinφ = np.sin(φ)
cosφ = np.cos(φ)
d = cosσ - coshτ
factor = self.scale_parameter * d**-2
return factor * np.array( # type: ignore
[
[cosφ * (cosσ * coshτ - 1), -cosφ * sinσ * sinhτ, sinφ * sinσ * d],
[sinφ * (cosσ * coshτ - 1), -sinφ * sinσ * sinhτ, -cosφ * sinσ * d],
[-sinσ * sinhτ, 1 - cosσ * coshτ, np.zeros_like(σ)],
]
)
def _volume_factor(self, points: np.ndarray) -> ArrayLike:
σ, τ = points[..., 0], points[..., 1]
return self.scale_parameter**3 * np.sin(σ) * (np.cosh(τ) - np.cos(σ)) ** -3 # type: ignore
def _scale_factors(self, points: np.ndarray) -> np.ndarray:
σ, τ = points[..., 0], points[..., 1]
sf = self.scale_parameter / (np.cosh(τ) - np.cos(σ))
return np.array([sf, sf, sf * np.sin(σ)])
def _basis_rotation(self, points: np.ndarray) -> np.ndarray:
σ, τ, φ = points[..., 0], points[..., 1], points[..., 2]
sinσ = np.sin(σ)
cosσ = np.cos(σ)
sinhτ = np.sinh(τ)
coshτ = np.cosh(τ)
sinφ = np.sin(φ)
cosφ = np.cos(φ)
d = cosσ - coshτ
return np.array(
[
[
cosφ * (1 - cosσ * coshτ) / d,
sinφ * (1 - cosσ * coshτ) / d,
sinσ * sinhτ / d,
],
[
cosφ * sinσ * sinhτ / d,
sinφ * sinσ * sinhτ / d,
(cosσ * coshτ - 1) / d,
],
[-sinφ, cosφ, np.zeros_like(σ)],
]
)