"""
.. 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 BipolarCoordinates(CoordinatesBase):
"""2-dimensional bipolar coordinates"""
dim = 2
axes = ["σ", "τ"]
_axes_alt = {"σ": ["sigma"], "τ": ["tau"]}
coordinate_limits = [(0, 2 * np.pi), (-np.inf, np.inf)]
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 = points[..., 0], points[..., 1]
denom = np.cosh(τ) - np.cos(σ)
x = self.scale_parameter * np.sinh(τ) / denom
y = self.scale_parameter * np.sin(σ) / denom
return np.stack((x, y), axis=-1)
def _pos_from_cart(self, points: np.ndarray) -> np.ndarray:
x, y = points[..., 0], points[..., 1]
a = self.scale_parameter
h2 = x**2 + y**2
denom = a**2 - h2 + np.sqrt((a**2 - h2) ** 2 + 4 * a**2 * y**2)
σ = np.mod(np.pi - 2 * np.arctan2(2 * a * y, denom), 2 * np.pi)
τ = 0.5 * np.log(((x + a) ** 2 + y**2) / ((x - a) ** 2 + y**2))
return np.stack((σ, τ), axis=-1)
def _mapping_jacobian(self, points: np.ndarray) -> np.ndarray:
σ, τ = points[..., 0], points[..., 1]
sinσ = np.sin(σ)
cosσ = np.cos(σ)
sinhτ = np.sinh(τ)
coshτ = np.cosh(τ)
factor = self.scale_parameter * (cosσ - coshτ) ** -2
return factor * np.array( # type: ignore
[
[-sinσ * sinhτ, 1 - cosσ * coshτ],
[cosσ * coshτ - 1, -sinσ * sinhτ],
]
)
def _volume_factor(self, points: np.ndarray) -> ArrayLike:
σ, τ = points[..., 0], points[..., 1]
return self.scale_parameter**2 * (np.cosh(τ) - np.cos(σ)) ** -2 # 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])
def _basis_rotation(self, points: np.ndarray) -> np.ndarray:
σ, τ = points[..., 0], points[..., 1]
sinσ = np.sin(σ)
cosσ = np.cos(σ)
sinhτ = np.sinh(τ)
coshτ = np.cosh(τ)
factor = 1 / (cosσ - coshτ)
return factor * np.array( # type: ignore
[
[sinσ * sinhτ, 1 - cosσ * coshτ],
[cosσ * coshτ - 1, sinσ * sinhτ],
]
)