"""
.. 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 SphericalCoordinates(CoordinatesBase):
"""3-dimensional spherical coordinates"""
dim = 3
axes = ["r", "θ", "φ"]
_axes_alt = {"θ": ["theta"], "φ": ["phi"]}
coordinate_limits = [(0, np.inf), (0, np.pi), (0, 2 * np.pi)]
major_axis = 0
_singleton: SphericalCoordinates | None = None
def __new__(cls):
# cache the instances for each dimension
if cls._singleton is None:
cls._singleton = super().__new__(cls)
return cls._singleton
def __repr__(self) -> str:
"""return instance as string"""
return f"{self.__class__.__name__}()"
def __eq__(self, other):
return self.__class__ is other.__class__
def _pos_to_cart(self, points: np.ndarray) -> np.ndarray:
r, θ, φ = points[..., 0], points[..., 1], points[..., 2]
rsinθ = r * np.sin(θ)
x = rsinθ * np.cos(φ)
y = rsinθ * np.sin(φ)
z = r * np.cos(θ)
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]
r = np.linalg.norm(points, axis=-1)
θ = np.arctan2(np.hypot(x, y), z)
φ = np.arctan2(y, x)
return np.stack((r, θ, φ), axis=-1)
def _mapping_jacobian(self, points: np.ndarray) -> np.ndarray:
r, θ, φ = points[..., 0], points[..., 1], points[..., 2]
sinθ, cosθ = np.sin(θ), np.cos(θ)
sinφ, cosφ = np.sin(φ), np.cos(φ)
return np.array(
[
[cosφ * sinθ, r * cosφ * cosθ, -r * sinφ * sinθ],
[sinφ * sinθ, r * sinφ * cosθ, r * cosφ * sinθ],
[cosθ, -r * sinθ, np.zeros_like(θ)],
]
)
def _volume_factor(self, points: np.ndarray) -> ArrayLike:
r, θ = points[..., 0], points[..., 1]
return r**2 * np.sin(θ) # type: ignore
def _cell_volume(self, c_low: np.ndarray, c_high: np.ndarray):
r1, θ1, φ1 = c_low[..., 0], c_low[..., 1], c_low[..., 2]
r2, θ2, φ2 = c_high[..., 0], c_high[..., 1], c_high[..., 2]
return (φ2 - φ1) * (np.cos(θ1) - np.cos(θ2)) * (r2**3 - r1**3) / 3
def _scale_factors(self, points: np.ndarray) -> np.ndarray:
r, θ = points[..., 0], points[..., 1]
return np.array([np.ones_like(r), r, r * np.sin(θ)])
def _basis_rotation(self, points: np.ndarray) -> np.ndarray:
θ, φ = points[..., 1], points[..., 2]
sinθ, cosθ = np.sin(θ), np.cos(θ)
sinφ, cosφ = np.sin(φ), np.cos(φ)
return np.array(
[
[cosφ * sinθ, sinφ * sinθ, cosθ],
[cosφ * cosθ, sinφ * cosθ, -sinθ],
[-sinφ, cosφ, np.zeros_like(θ)],
]
)