# Source code for astropy.modeling.rotations

```# Licensed under a 3-clause BSD style license - see LICENSE.rst

"""Implements rotations, including spherical rotations as defined in WCS Paper II [1]_.

`RotateNative2Celestial` and `RotateCelestial2Native` follow the convention in
WCS Paper II to rotate to/from a native sphere and the celestial sphere.

The implementation uses `EulerAngleRotation`. The model parameters are
three angles: the longitude (``lon``) and latitude (``lat``) of the fiducial point
in the celestial system (``CRVAL`` keywords in FITS), and the longitude of the celestial
pole in the native system (``lon_pole``). The Euler angles are ``lon+90``, ``90-lat``
and ``-(lon_pole-90)``.

References
----------
.. [1] Calabretta, M.R., Greisen, E.W., 2002, A&A, 395, 1077 (Paper II)
"""
# pylint: disable=invalid-name, too-many-arguments, no-member

import math
from functools import reduce

import numpy as np

from astropy import units as u
from astropy.coordinates.matrix_utilities import rotation_matrix

from .core import Model
from .parameters import Parameter
from .utils import _to_orig_unit, _to_radian

__all__ = [
"RotateCelestial2Native",
"RotateNative2Celestial",
"Rotation2D",
"EulerAngleRotation",
"RotationSequence3D",
"SphericalRotationSequence",
]

def _create_matrix(angles, axes_order):
matrices = []
for angle, axis in zip(angles, axes_order):
if isinstance(angle, u.Quantity):
angle = angle.value
angle = angle.item()
return reduce(np.matmul, matrices[::-1])

def spherical2cartesian(alpha, delta):
x = np.cos(alpha) * np.cos(delta)
y = np.cos(delta) * np.sin(alpha)
z = np.sin(delta)
return np.array([x, y, z])

def cartesian2spherical(x, y, z):
h = np.hypot(x, y)
alpha = np.rad2deg(np.arctan2(y, x))
delta = np.rad2deg(np.arctan2(z, h))
return alpha, delta

[docs]
class RotationSequence3D(Model):
"""
Perform a series of rotations about different axis in 3D space.

Positive angles represent a counter-clockwise rotation.

Parameters
----------
angles : array-like
Angles of rotation in deg in the order of axes_order.
axes_order : str
A sequence of 'x', 'y', 'z' corresponding to axis of rotation.

Examples
--------
>>> model = RotationSequence3D([1.1, 2.1, 3.1, 4.1], axes_order='xyzx')

"""

_separable = False
n_inputs = 3
n_outputs = 3

angles = Parameter(
default=[],
getter=_to_orig_unit,
description="Angles of rotation in deg in the order of axes_order",
)

def __init__(self, angles, axes_order, name=None):
self.axes = ["x", "y", "z"]
unrecognized = set(axes_order).difference(self.axes)
if unrecognized:
raise ValueError(
f"Unrecognized axis label {unrecognized}; should be one of {self.axes} "
)
self.axes_order = axes_order
if len(angles) != len(axes_order):
raise ValueError(
f"The number of angles {len(angles)} should match "
f"the number of axes {len(axes_order)}."
)
super().__init__(angles, name=name)
self._inputs = ("x", "y", "z")
self._outputs = ("x", "y", "z")

@property
def inverse(self):
"""Inverse rotation."""
angles = self.angles.value[::-1] * -1
return self.__class__(angles, axes_order=self.axes_order[::-1])

[docs]
def evaluate(self, x, y, z, angles):
"""
Apply the rotation to a set of 3D Cartesian coordinates.
"""
if x.shape != y.shape or x.shape != z.shape:
raise ValueError("Expected input arrays to have the same shape")
# Note: If the original shape was () (an array scalar) convert to a
# 1-element 1-D array on output for consistency with most other models
orig_shape = x.shape or (1,)
inarr = np.array([x.flatten(), y.flatten(), z.flatten()])
result = np.dot(_create_matrix(angles[0], self.axes_order), inarr)
x, y, z = result[0], result[1], result[2]
x.shape = y.shape = z.shape = orig_shape
return x, y, z

[docs]
class SphericalRotationSequence(RotationSequence3D):
"""
Perform a sequence of rotations about arbitrary number of axes
in spherical coordinates.

Parameters
----------
angles : list
A sequence of angles (in deg).
axes_order : str
A sequence of characters ('x', 'y', or 'z') corresponding to the
axis of rotation and matching the order in ``angles``.

"""

def __init__(self, angles, axes_order, name=None, **kwargs):
self._n_inputs = 2
self._n_outputs = 2
super().__init__(angles, axes_order=axes_order, name=name, **kwargs)
self._inputs = ("lon", "lat")
self._outputs = ("lon", "lat")

@property
def n_inputs(self):
return self._n_inputs

@property
def n_outputs(self):
return self._n_outputs

[docs]
def evaluate(self, lon, lat, angles):
x, y, z = spherical2cartesian(lon, lat)
x1, y1, z1 = super().evaluate(x, y, z, angles)
lon, lat = cartesian2spherical(x1, y1, z1)
return lon, lat

class _EulerRotation:
"""
Base class which does the actual computation.
"""

_separable = False

def evaluate(self, alpha, delta, phi, theta, psi, axes_order):
shape = None
if isinstance(alpha, np.ndarray):
alpha = alpha.flatten()
delta = delta.flatten()
shape = alpha.shape
inp = spherical2cartesian(alpha, delta)
matrix = _create_matrix([phi, theta, psi], axes_order)
result = np.dot(matrix, inp)
a, b = cartesian2spherical(*result)
if shape is not None:
a.shape = shape
b.shape = shape
return a, b

_input_units_strict = True

_input_units_allow_dimensionless = True

@property
def input_units(self):
"""Input units."""
return {self.inputs[0]: u.deg, self.inputs[1]: u.deg}

@property
def return_units(self):
"""Output units."""
return {self.outputs[0]: u.deg, self.outputs[1]: u.deg}

[docs]
class EulerAngleRotation(_EulerRotation, Model):
"""
Implements Euler angle intrinsic rotations.

Rotates one coordinate system into another (fixed) coordinate system.
All coordinate systems are right-handed. The sign of the angles is
determined by the right-hand rule..

Parameters
----------
phi, theta, psi : float or `~astropy.units.Quantity` ['angle']
"proper" Euler angles in deg.
If floats, they should be in deg.
axes_order : str
A 3 character string, a combination of 'x', 'y' and 'z',
where each character denotes an axis in 3D space.
"""

n_inputs = 2
n_outputs = 2

phi = Parameter(
default=0,
getter=_to_orig_unit,
description="1st Euler angle (Quantity or value in deg)",
)
theta = Parameter(
default=0,
getter=_to_orig_unit,
description="2nd Euler angle (Quantity or value in deg)",
)
psi = Parameter(
default=0,
getter=_to_orig_unit,
description="3rd Euler angle (Quantity or value in deg)",
)

def __init__(self, phi, theta, psi, axes_order, **kwargs):
self.axes = ["x", "y", "z"]
if len(axes_order) != 3:
raise TypeError(
"Expected axes_order to be a character sequence of length 3, "
f"got {axes_order}"
)
unrecognized = set(axes_order).difference(self.axes)
if unrecognized:
raise ValueError(
f"Unrecognized axis label {unrecognized}; should be one of {self.axes}"
)
self.axes_order = axes_order
qs = [isinstance(par, u.Quantity) for par in [phi, theta, psi]]
if any(qs) and not all(qs):
raise TypeError(
"All parameters should be of the same type - float or Quantity."
)

super().__init__(phi=phi, theta=theta, psi=psi, **kwargs)
self._inputs = ("alpha", "delta")
self._outputs = ("alpha", "delta")

@property
def inverse(self):
return self.__class__(
phi=-self.psi,
theta=-self.theta,
psi=-self.phi,
axes_order=self.axes_order[::-1],
)

[docs]
def evaluate(self, alpha, delta, phi, theta, psi):
a, b = super().evaluate(alpha, delta, phi, theta, psi, self.axes_order)
return a, b

class _SkyRotation(_EulerRotation, Model):
"""
Base class for RotateNative2Celestial and RotateCelestial2Native.
"""

lon = Parameter(
default=0, getter=_to_orig_unit, setter=_to_radian, description="Latitude"
)
lat = Parameter(
default=0, getter=_to_orig_unit, setter=_to_radian, description="Longtitude"
)
lon_pole = Parameter(
default=0,
getter=_to_orig_unit,
description="Longitude of a pole",
)

def __init__(self, lon, lat, lon_pole, **kwargs):
qs = [isinstance(par, u.Quantity) for par in [lon, lat, lon_pole]]
if any(qs) and not all(qs):
raise TypeError(
"All parameters should be of the same type - float or Quantity."
)
super().__init__(lon, lat, lon_pole, **kwargs)
self.axes_order = "zxz"

def _evaluate(self, phi, theta, lon, lat, lon_pole):
alpha, delta = super().evaluate(phi, theta, lon, lat, lon_pole, self.axes_order)
mask = alpha < 0
else:
alpha += 360
return alpha, delta

[docs]
class RotateNative2Celestial(_SkyRotation):
"""
Transform from Native to Celestial Spherical Coordinates.

Parameters
----------
lon : float or `~astropy.units.Quantity` ['angle']
Celestial longitude of the fiducial point.
lat : float or `~astropy.units.Quantity` ['angle']
Celestial latitude of the fiducial point.
lon_pole : float or `~astropy.units.Quantity` ['angle']
Longitude of the celestial pole in the native system.

Notes
-----
If ``lon``, ``lat`` and ``lon_pole`` are numerical values they
should be in units of deg. Inputs are angles on the native sphere.
Outputs are angles on the celestial sphere.
"""

n_inputs = 2
n_outputs = 2

@property
def input_units(self):
"""Input units."""
return {self.inputs[0]: u.deg, self.inputs[1]: u.deg}

@property
def return_units(self):
"""Output units."""
return {self.outputs[0]: u.deg, self.outputs[1]: u.deg}

def __init__(self, lon, lat, lon_pole, **kwargs):
super().__init__(lon, lat, lon_pole, **kwargs)
self.inputs = ("phi_N", "theta_N")
self.outputs = ("alpha_C", "delta_C")

[docs]
def evaluate(self, phi_N, theta_N, lon, lat, lon_pole):
"""
Parameters
----------
phi_N, theta_N : float or `~astropy.units.Quantity` ['angle']
Angles in the Native coordinate system.
it is assumed that numerical only inputs are in degrees.
If float, assumed in degrees.
lon, lat, lon_pole : float or `~astropy.units.Quantity` ['angle']
Parameter values when the model was initialized.
If float, assumed in degrees.

Returns
-------
alpha_C, delta_C : float or `~astropy.units.Quantity` ['angle']
Angles on the Celestial sphere.
If float, in degrees.
"""
# The values are in radians since they have already been through the setter.
if isinstance(lon, u.Quantity):
lon = lon.value
lat = lat.value
lon_pole = lon_pole.value
# Convert to Euler angles
phi = lon_pole - np.pi / 2
theta = -(np.pi / 2 - lat)
psi = -(np.pi / 2 + lon)
alpha_C, delta_C = super()._evaluate(phi_N, theta_N, phi, theta, psi)
return alpha_C, delta_C

@property
def inverse(self):
# convert to angles on the celestial sphere
return RotateCelestial2Native(self.lon, self.lat, self.lon_pole)

[docs]
class RotateCelestial2Native(_SkyRotation):
"""
Transform from Celestial to Native Spherical Coordinates.

Parameters
----------
lon : float or `~astropy.units.Quantity` ['angle']
Celestial longitude of the fiducial point.
lat : float or `~astropy.units.Quantity` ['angle']
Celestial latitude of the fiducial point.
lon_pole : float or `~astropy.units.Quantity` ['angle']
Longitude of the celestial pole in the native system.

Notes
-----
If ``lon``, ``lat`` and ``lon_pole`` are numerical values they should be
in units of deg. Inputs are angles on the celestial sphere.
Outputs are angles on the native sphere.
"""

n_inputs = 2
n_outputs = 2

@property
def input_units(self):
"""Input units."""
return {self.inputs[0]: u.deg, self.inputs[1]: u.deg}

@property
def return_units(self):
"""Output units."""
return {self.outputs[0]: u.deg, self.outputs[1]: u.deg}

def __init__(self, lon, lat, lon_pole, **kwargs):
super().__init__(lon, lat, lon_pole, **kwargs)

# Inputs are angles on the celestial sphere
self.inputs = ("alpha_C", "delta_C")
# Outputs are angles on the native sphere
self.outputs = ("phi_N", "theta_N")

[docs]
def evaluate(self, alpha_C, delta_C, lon, lat, lon_pole):
"""
Parameters
----------
alpha_C, delta_C : float or `~astropy.units.Quantity` ['angle']
Angles in the Celestial coordinate frame.
If float, assumed in degrees.
lon, lat, lon_pole : float or `~astropy.units.Quantity` ['angle']
Parameter values when the model was initialized.
If float, assumed in degrees.

Returns
-------
phi_N, theta_N : float or `~astropy.units.Quantity` ['angle']
Angles on the Native sphere.
If float, in degrees.

"""
if isinstance(lon, u.Quantity):
lon = lon.value
lat = lat.value
lon_pole = lon_pole.value
# Convert to Euler angles
phi = np.pi / 2 + lon
theta = np.pi / 2 - lat
psi = -(lon_pole - np.pi / 2)
phi_N, theta_N = super()._evaluate(alpha_C, delta_C, phi, theta, psi)

return phi_N, theta_N

@property
def inverse(self):
return RotateNative2Celestial(self.lon, self.lat, self.lon_pole)

[docs]
class Rotation2D(Model):
"""
Perform a 2D rotation given an angle.

Positive angles represent a counter-clockwise rotation and vice-versa.

Parameters
----------
angle : float or `~astropy.units.Quantity` ['angle']
Angle of rotation (if float it should be in deg).
"""

n_inputs = 2
n_outputs = 2

_separable = False

angle = Parameter(
default=0.0,
getter=_to_orig_unit,
description="Angle of rotation (Quantity or value in deg)",
)

def __init__(self, angle=angle, **kwargs):
super().__init__(angle=angle, **kwargs)
self._inputs = ("x", "y")
self._outputs = ("x", "y")

@property
def inverse(self):
"""Inverse rotation."""
return self.__class__(angle=-self.angle)

[docs]
@classmethod
def evaluate(cls, x, y, angle):
"""
Rotate (x, y) about ``angle``.

Parameters
----------
x, y : array-like
Input quantities
angle : float or `~astropy.units.Quantity` ['angle']
Angle of rotations.
If float, assumed in degrees.

"""
if x.shape != y.shape:
raise ValueError("Expected input arrays to have the same shape")

# If one argument has units, enforce they both have units and they are compatible.
x_unit = getattr(x, "unit", None)
y_unit = getattr(y, "unit", None)
has_units = x_unit is not None and y_unit is not None
if x_unit != y_unit:
if has_units and y_unit.is_equivalent(x_unit):
y = y.to(x_unit)
y_unit = x_unit
else:
raise u.UnitsError("x and y must have compatible units")

# Note: If the original shape was () (an array scalar) convert to a
# 1-element 1-D array on output for consistency with most other models
orig_shape = x.shape or (1,)
inarr = np.array([x.flatten(), y.flatten()])
if isinstance(angle, u.Quantity):
result = np.dot(cls._compute_matrix(angle), inarr)
x, y = result[0], result[1]
x.shape = y.shape = orig_shape
if has_units:
return u.Quantity(x, unit=x_unit, subok=True), u.Quantity(
y, unit=y_unit, subok=True
)
return x, y

@staticmethod
def _compute_matrix(angle):
if not np.isscalar(angle):
angle = angle[0]
return np.array(
[[math.cos(angle), -math.sin(angle)], [math.sin(angle), math.cos(angle)]],
dtype=np.float64,
)

```