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RotateAndTiltWrapperPotential.py
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RotateAndTiltWrapperPotential.py
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###############################################################################
# RotateAndTiltWrapperPotential.py: Wrapper to rotate and tilt the z-axis
# of a potential
###############################################################################
import numpy
from ..util import _rotate_to_arbitrary_vector, conversion, coords
from .Potential import (
_evaluatephitorques,
_evaluatePotentials,
_evaluateRforces,
_evaluatezforces,
check_potential_inputs_not_arrays,
evaluateDensities,
evaluatephi2derivs,
evaluatephizderivs,
evaluateR2derivs,
evaluateRphiderivs,
evaluateRzderivs,
evaluatez2derivs,
)
from .WrapperPotential import WrapperPotential
# Only implement 3D wrapper
class RotateAndTiltWrapperPotential(WrapperPotential):
"""Potential wrapper that allows a potential to be rotated in 3D
according to three orientation angles. These angles can either be
specified using:
* A rotation around the original z-axis (`galaxy_pa`) and the new direction of the z-axis (`zvec`) or
* A rotation around the original z-axis (`galaxy_pa`), the `inclination`, and a rotation around the new z axis (`sky_pa`).
The second option allows one to specify the inclination and sky position angle (measured from North) in the usual manner in extragalactic observations.
A final `offset` option allows one to apply a static offset in Cartesian coordinate space to be applied to the potential following the rotation and tilt.
"""
def __init__(
self,
amp=1.0,
inclination=None,
galaxy_pa=None,
sky_pa=None,
zvec=None,
offset=None,
pot=None,
ro=None,
vo=None,
):
"""
A potential that rotates and tilts another potential.
Parameters
----------
amp : float, optional
Overall amplitude to apply to the potential. Default is 1.0.
inclination : float or Quantity, optional
Usual inclination angle (with the line-of-sight being the z axis).
galaxy_pa : float or Quantity, optional
Rotation angle of the original potential around the original z axis.
sky_pa : float or Quantity, optional
Rotation angle around the inclined z axis (usual sky position angle measured from North).
zvec : numpy.ndarray, optional
3D vector specifying the direction of the rotated z axis.
offset : numpy.ndarray or Quantity, optional
Static offset in Cartesian coordinates.
pot : Potential instance or list thereof
The Potential instance or list thereof to rotate and tilt.
ro : float or Quantity, optional
Distance scale for translation into internal units (default from configuration file).
vo : float or Quantity, optional
Velocity scale for translation into internal units (default from configuration file).
Notes
-----
- 2021-03-29 - Started - Mackereth (UofT)
- 2021-04-18 - Added inclination, sky_pa, galaxy_pa setup - Bovy (UofT)
- 2022-03-14 - added offset kwarg - Mackereth (UofT)
"""
WrapperPotential.__init__(self, amp=amp, pot=pot, ro=ro, vo=vo, _init=True)
inclination = conversion.parse_angle(inclination)
sky_pa = conversion.parse_angle(sky_pa)
galaxy_pa = conversion.parse_angle(galaxy_pa)
zvec, galaxy_pa = self._parse_inclination(inclination, sky_pa, zvec, galaxy_pa)
self._offset = conversion.parse_length(
numpy.array(offset) if isinstance(offset, list) else offset, ro=self._ro
)
self._setup_zvec_pa(zvec, galaxy_pa)
self._norot = False
if (self._rot == numpy.eye(3)).all():
self._norot = True
self.hasC = True
self.hasC_dxdv = True
self.isNonAxi = True
def _parse_inclination(self, inclination, sky_pa, zvec, galaxy_pa):
if inclination is None:
return (zvec, galaxy_pa)
if sky_pa is None:
sky_pa = 0.0
zvec_rot = numpy.dot(
numpy.array(
[
[numpy.sin(sky_pa), numpy.cos(sky_pa), 0.0],
[-numpy.cos(sky_pa), numpy.sin(sky_pa), 0.0],
[0.0, 0.0, 1],
]
),
numpy.array(
[
[1.0, 0.0, 0.0],
[0.0, -numpy.cos(inclination), -numpy.sin(inclination)],
[0.0, -numpy.sin(inclination), numpy.cos(inclination)],
]
),
)
zvec = numpy.dot(zvec_rot, numpy.array([0.0, 0.0, 1.0]))
int_rot = _rotate_to_arbitrary_vector(
numpy.array([[0.0, 0.0, 1.0]]), zvec, inv=False
)[0]
pa = numpy.dot(int_rot, numpy.dot(zvec_rot, [1.0, 0.0, 0.0]))
return (zvec, galaxy_pa + numpy.arctan2(pa[1], pa[0]))
def _setup_zvec_pa(self, zvec, pa):
if not pa is None:
pa_rot = numpy.array(
[
[numpy.cos(pa), numpy.sin(pa), 0.0],
[-numpy.sin(pa), numpy.cos(pa), 0.0],
[0.0, 0.0, 1.0],
]
)
else:
pa_rot = numpy.eye(3)
if not zvec is None:
if not isinstance(zvec, numpy.ndarray):
zvec = numpy.array(zvec)
zvec /= numpy.sqrt(numpy.sum(zvec**2.0))
zvec_rot = _rotate_to_arbitrary_vector(
numpy.array([[0.0, 0.0, 1.0]]), zvec, inv=True
)[0]
else:
zvec_rot = numpy.eye(3)
self._rot = numpy.dot(pa_rot, zvec_rot)
self._inv_rot = numpy.linalg.inv(self._rot)
return None
def __getattr__(self, attribute):
return super().__getattr__(attribute)
@check_potential_inputs_not_arrays
def _evaluate(self, R, z, phi=0.0, t=0.0):
x, y, z = coords.cyl_to_rect(R, phi, z) if not numpy.isinf(R) else (R, 0.0, z)
if self._norot:
xyzp = numpy.array([x, y, z])
else:
xyzp = numpy.dot(self._rot, numpy.array([x, y, z]))
if self._offset is not None:
xyzp += self._offset
Rp, phip, zp = coords.rect_to_cyl(xyzp[0], xyzp[1], xyzp[2])
return _evaluatePotentials(self._pot, Rp, zp, phi=phip, t=t)
@check_potential_inputs_not_arrays
def _Rforce(self, R, z, phi=0.0, t=0.0):
Fxyz = self._force_xyz(R, z, phi=phi, t=t)
return numpy.cos(phi) * Fxyz[0] + numpy.sin(phi) * Fxyz[1]
@check_potential_inputs_not_arrays
def _phitorque(self, R, z, phi=0.0, t=0.0):
Fxyz = self._force_xyz(R, z, phi=phi, t=t)
return R * (-numpy.sin(phi) * Fxyz[0] + numpy.cos(phi) * Fxyz[1])
@check_potential_inputs_not_arrays
def _zforce(self, R, z, phi=0.0, t=0.0):
return self._force_xyz(R, z, phi=phi, t=t)[2]
def _force_xyz(self, R, z, phi=0.0, t=0.0):
"""Get the rectangular forces in the transformed frame"""
x, y, z = coords.cyl_to_rect(R, phi, z)
if self._norot:
xyzp = numpy.array([x, y, z])
else:
xyzp = numpy.dot(self._rot, numpy.array([x, y, z]))
if self._offset is not None:
xyzp += self._offset
Rp, phip, zp = coords.rect_to_cyl(xyzp[0], xyzp[1], xyzp[2])
Rforcep = _evaluateRforces(self._pot, Rp, zp, phi=phip, t=t)
phitorquep = _evaluatephitorques(self._pot, Rp, zp, phi=phip, t=t)
zforcep = _evaluatezforces(self._pot, Rp, zp, phi=phip, t=t)
xforcep = numpy.cos(phip) * Rforcep - numpy.sin(phip) * phitorquep / Rp
yforcep = numpy.sin(phip) * Rforcep + numpy.cos(phip) * phitorquep / Rp
return numpy.dot(self._inv_rot, numpy.array([xforcep, yforcep, zforcep]))
@check_potential_inputs_not_arrays
def _R2deriv(self, R, z, phi=0.0, t=0.0):
phi2 = self._2ndderiv_xyz(R, z, phi=phi, t=t)
return (
numpy.cos(phi) ** 2.0 * phi2[0, 0]
+ numpy.sin(phi) ** 2.0 * phi2[1, 1]
+ 2.0 * numpy.cos(phi) * numpy.sin(phi) * phi2[0, 1]
)
@check_potential_inputs_not_arrays
def _Rzderiv(self, R, z, phi=0.0, t=0.0):
phi2 = self._2ndderiv_xyz(R, z, phi=phi, t=t)
return numpy.cos(phi) * phi2[0, 2] + numpy.sin(phi) * phi2[1, 2]
@check_potential_inputs_not_arrays
def _z2deriv(self, R, z, phi=0.0, t=0.0):
return self._2ndderiv_xyz(R, z, phi=phi, t=t)[2, 2]
@check_potential_inputs_not_arrays
def _phi2deriv(self, R, z, phi=0.0, t=0.0):
Fxyz = self._force_xyz(R, z, phi=phi, t=t)
phi2 = self._2ndderiv_xyz(R, z, phi=phi, t=t)
return R**2.0 * (
numpy.sin(phi) ** 2.0 * phi2[0, 0]
+ numpy.cos(phi) ** 2.0 * phi2[1, 1]
- 2.0 * numpy.cos(phi) * numpy.sin(phi) * phi2[0, 1]
) + R * (numpy.cos(phi) * Fxyz[0] + numpy.sin(phi) * Fxyz[1])
@check_potential_inputs_not_arrays
def _Rphideriv(self, R, z, phi=0.0, t=0.0):
Fxyz = self._force_xyz(R, z, phi=phi, t=t)
phi2 = self._2ndderiv_xyz(R, z, phi=phi, t=t)
return (
R * numpy.cos(phi) * numpy.sin(phi) * (phi2[1, 1] - phi2[0, 0])
+ R * numpy.cos(2.0 * phi) * phi2[0, 1]
+ numpy.sin(phi) * Fxyz[0]
- numpy.cos(phi) * Fxyz[1]
)
@check_potential_inputs_not_arrays
def _phizderiv(self, R, z, phi=0.0, t=0.0):
phi2 = self._2ndderiv_xyz(R, z, phi=phi, t=t)
return R * (numpy.cos(phi) * phi2[1, 2] - numpy.sin(phi) * phi2[0, 2])
def _2ndderiv_xyz(self, R, z, phi=0.0, t=0.0):
"""Get the rectangular forces in the transformed frame"""
x, y, z = coords.cyl_to_rect(R, phi, z)
if self._norot:
xyzp = numpy.array([x, y, z])
else:
xyzp = numpy.dot(self._rot, numpy.array([x, y, z]))
if self._offset is not None:
xyzp += self._offset
Rp, phip, zp = coords.rect_to_cyl(xyzp[0], xyzp[1], xyzp[2])
Rforcep = _evaluateRforces(self._pot, Rp, zp, phi=phip, t=t)
phitorquep = _evaluatephitorques(self._pot, Rp, zp, phi=phip, t=t)
R2derivp = evaluateR2derivs(
self._pot, Rp, zp, phi=phip, t=t, use_physical=False
)
phi2derivp = evaluatephi2derivs(
self._pot, Rp, zp, phi=phip, t=t, use_physical=False
)
z2derivp = evaluatez2derivs(
self._pot, Rp, zp, phi=phip, t=t, use_physical=False
)
Rzderivp = evaluateRzderivs(
self._pot, Rp, zp, phi=phip, t=t, use_physical=False
)
Rphiderivp = evaluateRphiderivs(
self._pot, Rp, zp, phi=phip, t=t, use_physical=False
)
phizderivp = evaluatephizderivs(
self._pot, Rp, zp, phi=phip, t=t, use_physical=False
)
cp, sp = numpy.cos(phip), numpy.sin(phip)
cp2, sp2, cpsp = cp**2.0, sp**2.0, cp * sp
Rp2 = Rp * Rp
x2derivp = (
R2derivp * cp2
- 2.0 * Rphiderivp * cpsp / Rp
+ phi2derivp * sp2 / Rp2
- Rforcep * sp2 / Rp
- 2.0 * phitorquep * cpsp / Rp2
)
y2derivp = (
R2derivp * sp2
+ 2.0 * Rphiderivp * cpsp / Rp
+ phi2derivp * cp2 / Rp2
- Rforcep * cp2 / Rp
+ 2.0 * phitorquep * cpsp / Rp2
)
xyderivp = (
R2derivp * cpsp
+ Rphiderivp * (cp2 - sp2) / Rp
- phi2derivp * cpsp / Rp2
+ Rforcep * cpsp / Rp
+ phitorquep * (cp2 - sp2) / Rp2
)
xzderivp = Rzderivp * cp - phizderivp * sp / Rp
yzderivp = Rzderivp * sp + phizderivp * cp / Rp
return numpy.dot(
self._inv_rot,
numpy.dot(
numpy.array(
[
[x2derivp, xyderivp, xzderivp],
[xyderivp, y2derivp, yzderivp],
[xzderivp, yzderivp, z2derivp],
]
),
self._inv_rot.T,
),
)
@check_potential_inputs_not_arrays
def _dens(self, R, z, phi=0.0, t=0.0):
x, y, z = coords.cyl_to_rect(R, phi, z) if not numpy.isinf(R) else (R, 0.0, z)
if self._norot:
xyzp = numpy.array([x, y, z])
else:
xyzp = numpy.dot(self._rot, numpy.array([x, y, z]))
if self._offset is not None:
xyzp += self._offset
Rp, phip, zp = coords.rect_to_cyl(xyzp[0], xyzp[1], xyzp[2])
return evaluateDensities(self._pot, Rp, zp, phi=phip, t=t, use_physical=False)