"""Built-in Pisces coordinate systems."""
import numpy as np
from .base import CoordinateSystem
[docs]
class Cartesian1DCoordinateSystem(CoordinateSystem):
r"""
One-dimensional Cartesian coordinate system.
Coordinates
-----------
- :math:`x`: Position along a 1D Euclidean axis.
Conversion
----------
This system is already Cartesian. Conversion functions return the input.
.. math::
x = x
Notes
-----
This coordinate system is used in problems where only one spatial
dimension is relevant.
"""
__IS_ABSTRACT__ = False
__PARAMETERS__ = {}
__NDIM__ = 1
__AXES__ = ["x"]
[docs]
def convert_to_cartesian(self, x):
return x
[docs]
def convert_from_cartesian(self, x):
return x
[docs]
class Cartesian2DCoordinateSystem(CoordinateSystem):
r"""
Two-dimensional Cartesian coordinate system.
Coordinates
-----------
- :math:`x`: Horizontal position.
- :math:`y`: Vertical position.
Conversion
----------
This system is already Cartesian. Conversion functions return the input.
.. math::
x = x \\
y = y
Notes
-----
Cartesian 2D coordinates describe flat Euclidean planes with orthogonal axes.
"""
__IS_ABSTRACT__ = False
__PARAMETERS__ = {}
__NDIM__ = 2
__AXES__ = ["x", "y"]
[docs]
def convert_to_cartesian(self, x, y):
return x, y
[docs]
def convert_from_cartesian(self, x, y):
return x, y
[docs]
class Cartesian3DCoordinateSystem(CoordinateSystem):
r"""
Three-dimensional Cartesian coordinate system.
Coordinates
-----------
- :math:`x`: Position along the x-axis.
- :math:`y`: Position along the y-axis.
- :math:`z`: Position along the z-axis.
Conversion
----------
This system is already Cartesian. Conversion functions return the input.
.. math::
x = x \\
y = y \\
z = z
Notes
-----
Standard orthonormal coordinate system in 3D Euclidean space.
"""
__IS_ABSTRACT__ = False
__PARAMETERS__ = {}
__NDIM__ = 3
__AXES__ = ["x", "y", "z"]
[docs]
def convert_to_cartesian(self, x, y, z):
return x, y, z
[docs]
def convert_from_cartesian(self, x, y, z):
return x, y, z
[docs]
class SphericalCoordinateSystem(CoordinateSystem):
r"""
Spherical coordinate system in three dimensions.
Coordinates
-----------
- :math:`r`: Radial distance from the origin.
- :math:`\theta`: Polar angle (radians) from the positive z-axis.
- :math:`\phi`: Azimuthal angle (radians) from the positive x-axis.
Conversion to Cartesian
-----------------------
.. math::
x = r \sin\theta \cos\phi \\
y = r \sin\theta \sin\phi \\
z = r \cos\theta
Conversion from Cartesian
-------------------------
.. math::
r = \sqrt{x^2 + y^2 + z^2} \\
\theta = \arccos\left(\frac{z}{r}\right) \\
\phi = \arctan2(y, x)
Notes
-----
This system is used for problems with spherical symmetry. All angles
are expressed in radians.
"""
__IS_ABSTRACT__ = False
__PARAMETERS__ = {}
__NDIM__ = 3
__AXES__ = ["r", "theta", "phi"]
[docs]
def convert_to_cartesian(self, r, theta, phi):
# theta: polar angle (from z-axis), phi: azimuthal angle (from x-axis)
x = r * np.sin(theta) * np.cos(phi)
y = r * np.sin(theta) * np.sin(phi)
z = r * np.cos(theta)
return x, y, z
[docs]
def convert_from_cartesian(self, x, y, z):
r = np.sqrt(x**2 + y**2 + z**2)
theta = np.arccos(z / r)
phi = np.arctan2(y, x)
return r, theta, phi
[docs]
class CylindricalCoordinateSystem(CoordinateSystem):
r"""
Cylindrical coordinate system in three dimensions.
Coordinates
-----------
- :math:`r`: Radial distance from the z-axis.
- :math:`\theta`: Azimuthal angle (radians) from the x-axis.
- :math:`z`: Height along the z-axis.
Conversion to Cartesian
-----------------------
.. math::
x = r \cos\theta \\
y = r \sin\theta \\
z = z
Conversion from Cartesian
-------------------------
.. math::
r = \sqrt{x^2 + y^2} \\
\theta = \arctan2(y, x) \\
z = z
Notes
-----
Suitable for problems with axial or cylindrical symmetry.
"""
__IS_ABSTRACT__ = False
__PARAMETERS__ = {}
__NDIM__ = 3
__AXES__ = ["r", "theta", "z"]
[docs]
def convert_to_cartesian(self, r, theta, z):
x = r * np.cos(theta)
y = r * np.sin(theta)
return (x, y, z)
[docs]
def convert_from_cartesian(self, x, y, z):
r = np.hypot(x, y)
theta = np.arctan2(y, x)
return (r, theta, z)
[docs]
class PolarCoordinatesSystem(CoordinateSystem):
r"""
Polar coordinate system in two dimensions.
Coordinates
-----------
- :math:`r`: Radial distance from the origin.
- :math:`\theta`: Angle (radians) from the positive x-axis.
Conversion to Cartesian
-----------------------
.. math::
x = r \cos\theta \\
y = r \sin\theta
Conversion from Cartesian
-------------------------
.. math::
r = \sqrt{x^2 + y^2} \\
\theta = \arctan2(y, x)
Notes
-----
Commonly used for circular or rotational symmetry in 2D. Angles are in radians.
"""
__IS_ABSTRACT__ = False
__PARAMETERS__ = {}
__NDIM__ = 2
__AXES__ = ["r", "theta"]
[docs]
def convert_to_cartesian(self, r, theta):
x = r * np.cos(theta)
y = r * np.sin(theta)
return x, y
[docs]
def convert_from_cartesian(self, x, y):
r = np.hypot(x, y)
theta = np.arctan2(y, x)
return r, theta
