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)
"""

import math

import numpy as np

from .core import Model
from .parameters import Parameter
from astropy.coordinates.matrix_utilities import rotation_matrix, matrix_product
from astropy import units as u
from .utils import _to_radian, _to_orig_unit

__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()
        matrices.append(rotation_matrix(angle, axis, unit=u.rad))
    result = matrix_product(*matrices[::-1])
    return result


def spherical2cartesian(alpha, delta):
    alpha = np.deg2rad(alpha)
    delta = np.deg2rad(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') """ standard_broadcasting = False _separable = False n_inputs = 3 n_outputs = 3 angles = Parameter(default=[], getter=_to_orig_unit, setter=_to_radian) def __init__(self, angles, axes_order, name=None): self.axes = ['x', 'y', 'z'] unrecognized = set(axes_order).difference(self.axes) if unrecognized: raise ValueError("Unrecognized axis label {0}; " "should be one of {1} ".format(unrecognized, self.axes)) self.axes_order = axes_order if len(angles) != len(axes_order): raise ValueError("The number of angles {0} should match the number \ of axes {1}.".format(len(angles), 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 != 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) and alpha.ndim == 2: 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 {'alpha': u.deg, 'delta': u.deg} @property def return_units(self): """ Output units. """ return {'alpha': u.deg, 'delta': 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` "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, setter=_to_radian) theta = Parameter(default=0, getter=_to_orig_unit, setter=_to_radian) psi = Parameter(default=0, getter=_to_orig_unit, setter=_to_radian) 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," "got {}".format(axes_order)) unrecognized = set(axes_order).difference(self.axes) if unrecognized: raise ValueError("Unrecognized axis label {}; " "should be one of {} ".format(unrecognized, 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') 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) lat = Parameter(default=0, getter=_to_orig_unit, setter=_to_radian) lon_pole = Parameter(default=0, getter=_to_orig_unit, setter=_to_radian) 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 if isinstance(mask, np.ndarray): alpha[mask] += 360 else: alpha += 360 return alpha, delta
[docs]class RotateNative2Celestial(_SkyRotation): """ Transform from Native to Celestial Spherical Coordinates. Parameters ---------- lon : float or or `~astropy.units.Quantity` Celestial longitude of the fiducial point. lat : float or or `~astropy.units.Quantity` Celestial latitude of the fiducial point. lon_pole : float or or `~astropy.units.Quantity` 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 {'phi_N': u.deg, 'theta_N': u.deg} @property def return_units(self): """ Output units. """ return {'alpha_C': u.deg, 'delta_C': 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 (deg) or `~astropy.units.Quantity` Angles in the Native coordinate system. lon, lat, lon_pole : float (in deg) or `~astropy.units.Quantity` Parameter values when the model was initialized. Returns ------- alpha_C, delta_C : float (deg) or `~astropy.units.Quantity` Angles on the Celestial sphere. """ # 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 or `~astropy.units.Quantity` Celestial longitude of the fiducial point. lat : float or or `~astropy.units.Quantity` Celestial latitude of the fiducial point. lon_pole : float or or `~astropy.units.Quantity` 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 {'alpha_C': u.deg, 'delta_C': u.deg} @property def return_units(self): """ Output units. """ return {'phi_N': u.deg, 'theta_N': 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 (deg) or `~astropy.units.Quantity` Angles in the Celestial coordinate frame. lon, lat, lon_pole : float (deg) or `~astropy.units.Quantity` Parameter values when the model was initialized. Returns ------- phi_N, theta_N : float (deg) or `~astropy.units.Quantity` Angles on the Native sphere. """ 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 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, setter=_to_radian) 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 (deg) or `~astropy.units.Quantity` Angle of rotations. """ 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): angle = angle.to_value(u.rad) 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), u.Quantity(y, unit=y_unit) else: return x, y
@staticmethod def _compute_matrix(angle): return np.array([[math.cos(angle), -math.sin(angle)], [math.sin(angle), math.cos(angle)]], dtype=np.float64)