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# Source code for astropy.coordinates.representation.spherical

    # Licensed under a 3-clause BSD style license - see LICENSE.rst
    """Spherical representations and differentials."""

    import operator

    import numpy as np
    from erfa import ufunc as erfa_ufunc

    import astropy.units as u
    from astropy.coordinates.angles import Angle, Latitude, Longitude
    from astropy.coordinates.distances import Distance
    from astropy.coordinates.matrix_utilities import is_O3
    from astropy.utils import classproperty
    from astropy.utils.compat import COPY_IF_NEEDED

    from .base import BaseDifferential, BaseRepresentation
    from .cartesian import CartesianRepresentation

    class UnitSphericalRepresentation(BaseRepresentation):
        """
        Representation of points on a unit sphere.

        Parameters
        ----------
        lon, lat : `~astropy.units.Quantity` ['angle'] or str
            The longitude and latitude of the point(s), in angular units. The
            latitude should be between -90 and 90 degrees, and the longitude will
            be wrapped to an angle between 0 and 360 degrees. These can also be
            instances of `~astropy.coordinates.Angle`,
            `~astropy.coordinates.Longitude`, or `~astropy.coordinates.Latitude`.

        differentials : dict, `~astropy.coordinates.BaseDifferential`, optional
            Any differential classes that should be associated with this
            representation. The input must either be a single `~astropy.coordinates.BaseDifferential`
            instance (see `._compatible_differentials` for valid types), or a
            dictionary of of differential instances with keys set to a string
            representation of the SI unit with which the differential (derivative)
            is taken. For example, for a velocity differential on a positional
            representation, the key would be ``'s'`` for seconds, indicating that
            the derivative is a time derivative.

        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        attr_classes = {"lon": Longitude, "lat": Latitude}

        @classproperty
        def _dimensional_representation(cls):
            return SphericalRepresentation

        def __init__(self, lon, lat=None, differentials=None, copy=True):
            super().__init__(lon, lat, differentials=differentials, copy=copy)

        @classproperty
        def _compatible_differentials(cls):
            return [
                UnitSphericalDifferential,
                UnitSphericalCosLatDifferential,
                SphericalDifferential,
                SphericalCosLatDifferential,
                RadialDifferential,
            ]

        # Could let the metaclass define these automatically, but good to have
        # a bit clearer docstrings.
        @property
        def lon(self):
            """
            The longitude of the point(s).
            """
            return self._lon

        @property
        def lat(self):
            """
            The latitude of the point(s).
            """
            return self._lat

        def unit_vectors(self):
            sinlon, coslon = np.sin(self.lon), np.cos(self.lon)
            sinlat, coslat = np.sin(self.lat), np.cos(self.lat)
            return {
                "lon": CartesianRepresentation(-sinlon, coslon, 0.0, copy=COPY_IF_NEEDED),
                "lat": CartesianRepresentation(
                    -sinlat * coslon, -sinlat * sinlon, coslat, copy=COPY_IF_NEEDED
                ),
            }

        def scale_factors(self, omit_coslat=False):
            sf_lat = np.broadcast_to(1.0 / u.radian, self.shape, subok=True)
            sf_lon = sf_lat if omit_coslat else np.cos(self.lat) / u.radian
            return {"lon": sf_lon, "lat": sf_lat}

        def to_cartesian(self):
            """
            Converts spherical polar coordinates to 3D rectangular cartesian
            coordinates.
            """
            # erfa s2c: Convert [unit]spherical coordinates to Cartesian.
            p = erfa_ufunc.s2c(self.lon, self.lat)
            return CartesianRepresentation(p, xyz_axis=-1, copy=False)

        @classmethod
        def from_cartesian(cls, cart):
            """
            Converts 3D rectangular cartesian coordinates to spherical polar
            coordinates.
            """
            p = cart.get_xyz(xyz_axis=-1)
            # erfa c2s: P-vector to [unit]spherical coordinates.
            return cls(*erfa_ufunc.c2s(p), copy=False)

        def represent_as(self, other_class, differential_class=None):
            # Take a short cut if the other class is a spherical representation
            # TODO! for differential_class. This cannot (currently) be implemented
            # like in the other Representations since `_re_represent_differentials`
            # keeps differentials' unit keys, but this can result in a mismatch
            # between the UnitSpherical expected key (e.g. "s") and that expected
            # in the other class (here "s / m"). For more info, see PR #11467
            if isinstance(other_class, type) and not differential_class:
                if issubclass(other_class, PhysicsSphericalRepresentation):
                    return other_class(
                        phi=self.lon,
                        theta=90 * u.deg - self.lat,
                        r=1.0,
                        copy=COPY_IF_NEEDED,
                    )
                elif issubclass(other_class, SphericalRepresentation):
                    return other_class(
                        lon=self.lon,
                        lat=self.lat,
                        distance=1.0,
                        copy=COPY_IF_NEEDED,
                    )

            return super().represent_as(other_class, differential_class)

        def transform(self, matrix):
            r"""Transform the unit-spherical coordinates using a 3x3 matrix.

            This returns a new representation and does not modify the original one.
            Any differentials attached to this representation will also be
            transformed.

            Parameters
            ----------
            matrix : (3,3) array-like
                A 3x3 matrix, such as a rotation matrix (or a stack of matrices).

            Returns
            -------
            `~astropy.coordinates.UnitSphericalRepresentation` or `~astropy.coordinates.SphericalRepresentation`
                If ``matrix`` is O(3) -- :math:`M \dot M^T = I` -- like a rotation,
                then the result is a `~astropy.coordinates.UnitSphericalRepresentation`.
                All other matrices will change the distance, so the dimensional
                representation is used instead.

            """
            # the transformation matrix does not need to be a rotation matrix,
            # so the unit-distance is not guaranteed. For speed, we check if the
            # matrix is in O(3) and preserves lengths.
            if np.all(is_O3(matrix)):  # remain in unit-rep
                xyz = erfa_ufunc.s2c(self.lon, self.lat)
                p = erfa_ufunc.rxp(matrix, xyz)
                lon, lat = erfa_ufunc.c2s(p)
                rep = self.__class__(lon=lon, lat=lat)
                # handle differentials
                new_diffs = {
                    k: d.transform(matrix, self, rep) for k, d in self.differentials.items()
                }
                rep = rep.with_differentials(new_diffs)

            else:  # switch to dimensional representation
                rep = self._dimensional_representation(
                    lon=self.lon, lat=self.lat, distance=1, differentials=self.differentials
                ).transform(matrix)

            return rep

        def _scale_operation(self, op, *args):
            return self._dimensional_representation(
                lon=self.lon, lat=self.lat, distance=1.0, differentials=self.differentials
            )._scale_operation(op, *args)

        def __neg__(self):
            if any(
                differential.base_representation is not self.__class__
                for differential in self.differentials.values()
            ):
                return super().__neg__()

            result = self.__class__(self.lon + 180.0 * u.deg, -self.lat, copy=False)
            for key, differential in self.differentials.items():
                new_comps = (
                    op(getattr(differential, comp))
                    for op, comp in zip(
                        (operator.pos, operator.neg), differential.components
                    )
                )
                result.differentials[key] = differential.__class__(*new_comps, copy=False)
            return result

        def norm(self):
            """Vector norm.

            The norm is the standard Frobenius norm, i.e., the square root of the
            sum of the squares of all components with non-angular units, which is
            always unity for vectors on the unit sphere.

            Returns
            -------
            norm : `~astropy.units.Quantity` ['dimensionless']
                Dimensionless ones, with the same shape as the representation.
            """
            return u.Quantity(np.ones(self.shape), u.dimensionless_unscaled, copy=False)

        def _combine_operation(self, op, other, reverse=False):
            self._raise_if_has_differentials(op.__name__)

            result = self.to_cartesian()._combine_operation(op, other, reverse)
            if result is NotImplemented:
                return NotImplemented
            else:
                return self._dimensional_representation.from_cartesian(result)

        def mean(self, *args, **kwargs):
            """Vector mean.

            The representation is converted to cartesian, the means of the x, y,
            and z components are calculated, and the result is converted to a
            `~astropy.coordinates.SphericalRepresentation`.

            Refer to `~numpy.mean` for full documentation of the arguments, noting
            that ``axis`` is the entry in the ``shape`` of the representation, and
            that the ``out`` argument cannot be used.
            """
            self._raise_if_has_differentials("mean")
            return self._dimensional_representation.from_cartesian(
                self.to_cartesian().mean(*args, **kwargs)
            )

        def sum(self, *args, **kwargs):
            """Vector sum.

            The representation is converted to cartesian, the sums of the x, y,
            and z components are calculated, and the result is converted to a
            `~astropy.coordinates.SphericalRepresentation`.

            Refer to `~numpy.sum` for full documentation of the arguments, noting
            that ``axis`` is the entry in the ``shape`` of the representation, and
            that the ``out`` argument cannot be used.
            """
            self._raise_if_has_differentials("sum")
            return self._dimensional_representation.from_cartesian(
                self.to_cartesian().sum(*args, **kwargs)
            )

        def cross(self, other):
            """Cross product of two representations.

            The calculation is done by converting both ``self`` and ``other``
            to `~astropy.coordinates.CartesianRepresentation`, and converting the
            result back to `~astropy.coordinates.SphericalRepresentation`.

            Parameters
            ----------
            other : `~astropy.coordinates.BaseRepresentation` subclass instance
                The representation to take the cross product with.

            Returns
            -------
            cross_product : `~astropy.coordinates.SphericalRepresentation`
                With vectors perpendicular to both ``self`` and ``other``.
            """
            self._raise_if_has_differentials("cross")
            return self._dimensional_representation.from_cartesian(
                self.to_cartesian().cross(other)
            )

    class RadialRepresentation(BaseRepresentation):
        """
        Representation of the distance of points from the origin.

        Note that this is mostly intended as an internal helper representation.
        It can do little else but being used as a scale in multiplication.

        Parameters
        ----------
        distance : `~astropy.units.Quantity` ['length']
            The distance of the point(s) from the origin.

        differentials : dict, `~astropy.coordinates.BaseDifferential`, optional
            Any differential classes that should be associated with this
            representation. The input must either be a single `~astropy.coordinates.BaseDifferential`
            instance (see `._compatible_differentials` for valid types), or a
            dictionary of of differential instances with keys set to a string
            representation of the SI unit with which the differential (derivative)
            is taken. For example, for a velocity differential on a positional
            representation, the key would be ``'s'`` for seconds, indicating that
            the derivative is a time derivative.

        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        attr_classes = {"distance": u.Quantity}

        def __init__(self, distance, differentials=None, copy=True):
            super().__init__(distance, differentials=differentials, copy=copy)

        @property
        def distance(self):
            """
            The distance from the origin to the point(s).
            """
            return self._distance

        def unit_vectors(self):
            """Cartesian unit vectors are undefined for radial representation."""
            raise NotImplementedError(
                f"Cartesian unit vectors are undefined for {self.__class__} instances"
            )

        def scale_factors(self):
            l = np.broadcast_to(1.0 * u.one, self.shape, subok=True)
            return {"distance": l}

        def to_cartesian(self):
            """Cannot convert radial representation to cartesian."""
            raise NotImplementedError(
                f"cannot convert {self.__class__} instance to cartesian."
            )

        @classmethod
        def from_cartesian(cls, cart):
            """
            Converts 3D rectangular cartesian coordinates to radial coordinate.
            """
            return cls(distance=cart.norm(), copy=False)

        def __mul__(self, other):
            if isinstance(other, BaseRepresentation):
                return self.distance * other
            else:
                return super().__mul__(other)

        def norm(self):
            """Vector norm.

            Just the distance itself.

            Returns
            -------
            norm : `~astropy.units.Quantity` ['dimensionless']
                Dimensionless ones, with the same shape as the representation.
            """
            return self.distance

        def _combine_operation(self, op, other, reverse=False):
            return NotImplemented

        def transform(self, matrix):
            """Radial representations cannot be transformed by a Cartesian matrix.

            Parameters
            ----------
            matrix : array-like
                The transformation matrix in a Cartesian basis.
                Must be a multiplication: a diagonal matrix with identical elements.
                Must have shape (..., 3, 3), where the last 2 indices are for the
                matrix on each other axis. Make sure that the matrix shape is
                compatible with the shape of this representation.

            Raises
            ------
            ValueError
                If the matrix is not a multiplication.

            """
            scl = matrix[..., 0, 0]
            # check that the matrix is a scaled identity matrix on the last 2 axes.
            if np.any(matrix != scl[..., np.newaxis, np.newaxis] * np.identity(3)):
                raise ValueError(
                    "Radial representations can only be "
                    "transformed by a scaled identity matrix"
                )

            return self * scl

    def _spherical_op_funcs(op, *args):
        """For given operator, return functions that adjust lon, lat, distance."""
        if op is operator.neg:
            return lambda x: x + 180 * u.deg, operator.neg, operator.pos

        try:
            scale_sign = np.sign(args[0])
        except Exception:
            # This should always work, even if perhaps we get a negative distance.
            return operator.pos, operator.pos, lambda x: op(x, *args)

        scale = abs(args[0])
        return (
            lambda x: x + 180 * u.deg * np.signbit(scale_sign),
            lambda x: x * scale_sign,
            lambda x: op(x, scale),
        )

    class SphericalRepresentation(BaseRepresentation):
        """
        Representation of points in 3D spherical coordinates.

        Parameters
        ----------
        lon, lat : `~astropy.units.Quantity` ['angle']
            The longitude and latitude of the point(s), in angular units. The
            latitude should be between -90 and 90 degrees, and the longitude will
            be wrapped to an angle between 0 and 360 degrees. These can also be
            instances of `~astropy.coordinates.Angle`,
            `~astropy.coordinates.Longitude`, or `~astropy.coordinates.Latitude`.

        distance : `~astropy.units.Quantity` ['length']
            The distance to the point(s). If the distance is a length, it is
            passed to the :class:`~astropy.coordinates.Distance` class, otherwise
            it is passed to the :class:`~astropy.units.Quantity` class.

        differentials : dict, `~astropy.coordinates.BaseDifferential`, optional
            Any differential classes that should be associated with this
            representation. The input must either be a single `~astropy.coordinates.BaseDifferential`
            instance (see `._compatible_differentials` for valid types), or a
            dictionary of of differential instances with keys set to a string
            representation of the SI unit with which the differential (derivative)
            is taken. For example, for a velocity differential on a positional
            representation, the key would be ``'s'`` for seconds, indicating that
            the derivative is a time derivative.

        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        attr_classes = {"lon": Longitude, "lat": Latitude, "distance": u.Quantity}
        _unit_representation = UnitSphericalRepresentation

        def __init__(self, lon, lat=None, distance=None, differentials=None, copy=True):
            super().__init__(lon, lat, distance, copy=copy, differentials=differentials)
            if (
                not isinstance(self._distance, Distance)
                and self._distance.unit.physical_type == "length"
            ):
                try:
                    self._distance = Distance(self._distance, copy=False)
                except ValueError as e:
                    if e.args[0].startswith("distance must be >= 0"):
                        raise ValueError(
                            "Distance must be >= 0. To allow negative distance values, you"
                            " must explicitly pass in a `Distance` object with the "
                            "argument 'allow_negative=True'."
                        ) from e
                    raise

        @classproperty
        def _compatible_differentials(cls):
            return [
                UnitSphericalDifferential,
                UnitSphericalCosLatDifferential,
                SphericalDifferential,
                SphericalCosLatDifferential,
                RadialDifferential,
            ]

        @property
        def lon(self):
            """
            The longitude of the point(s).
            """
            return self._lon

        @property
        def lat(self):
            """
            The latitude of the point(s).
            """
            return self._lat

        @property
        def distance(self):
            """
            The distance from the origin to the point(s).
            """
            return self._distance

        def unit_vectors(self):
            sinlon, coslon = np.sin(self.lon), np.cos(self.lon)
            sinlat, coslat = np.sin(self.lat), np.cos(self.lat)
            return {
                "lon": CartesianRepresentation(-sinlon, coslon, 0.0, copy=COPY_IF_NEEDED),
                "lat": CartesianRepresentation(
                    -sinlat * coslon, -sinlat * sinlon, coslat, copy=COPY_IF_NEEDED
                ),
                "distance": CartesianRepresentation(
                    coslat * coslon, coslat * sinlon, sinlat, copy=COPY_IF_NEEDED
                ),
            }

        def scale_factors(self, omit_coslat=False):
            sf_lat = self.distance / u.radian
            sf_lon = sf_lat if omit_coslat else sf_lat * np.cos(self.lat)
            sf_distance = np.broadcast_to(1.0 * u.one, self.shape, subok=True)
            return {"lon": sf_lon, "lat": sf_lat, "distance": sf_distance}

        def represent_as(self, other_class, differential_class=None):
            # Take a short cut if the other class is a spherical representation

            if isinstance(other_class, type):
                if issubclass(other_class, PhysicsSphericalRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        phi=self.lon,
                        theta=90 * u.deg - self.lat,
                        r=self.distance,
                        differentials=diffs,
                        copy=False,
                    )

                elif issubclass(other_class, UnitSphericalRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        lon=self.lon, lat=self.lat, differentials=diffs, copy=False
                    )

                elif issubclass(other_class, RadialRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        distance=self.distance,
                        differentials=diffs,
                        copy=False,
                    )

            return super().represent_as(other_class, differential_class)

        def to_cartesian(self):
            """
            Converts spherical polar coordinates to 3D rectangular cartesian
            coordinates.
            """
            # We need to convert Distance to Quantity to allow negative values.
            if isinstance(self.distance, Distance):
                d = self.distance.view(u.Quantity)
            else:
                d = self.distance

            # erfa s2p: Convert spherical polar coordinates to p-vector.
            p = erfa_ufunc.s2p(self.lon, self.lat, d)

            return CartesianRepresentation(p, xyz_axis=-1, copy=False)

        @classmethod
        def from_cartesian(cls, cart):
            """
            Converts 3D rectangular cartesian coordinates to spherical polar
            coordinates.
            """
            p = cart.get_xyz(xyz_axis=-1)
            # erfa p2s: P-vector to spherical polar coordinates.
            return cls(*erfa_ufunc.p2s(p), copy=False)

        def transform(self, matrix):
            """Transform the spherical coordinates using a 3x3 matrix.

            This returns a new representation and does not modify the original one.
            Any differentials attached to this representation will also be
            transformed.

            Parameters
            ----------
            matrix : (3,3) array-like
                A 3x3 matrix, such as a rotation matrix (or a stack of matrices).

            """
            xyz = erfa_ufunc.s2c(self.lon, self.lat)
            p = erfa_ufunc.rxp(matrix, xyz)
            lon, lat, ur = erfa_ufunc.p2s(p)
            rep = self.__class__(lon=lon, lat=lat, distance=self.distance * ur)

            # handle differentials
            new_diffs = {
                k: d.transform(matrix, self, rep) for k, d in self.differentials.items()
            }
            return rep.with_differentials(new_diffs)

        def norm(self):
            """Vector norm.

            The norm is the standard Frobenius norm, i.e., the square root of the
            sum of the squares of all components with non-angular units.  For
            spherical coordinates, this is just the absolute value of the distance.

            Returns
            -------
            norm : `astropy.units.Quantity`
                Vector norm, with the same shape as the representation.
            """
            return np.abs(self.distance)

        def _scale_operation(self, op, *args):
            # TODO: expand special-casing to UnitSpherical and RadialDifferential.
            if any(
                differential.base_representation is not self.__class__
                for differential in self.differentials.values()
            ):
                return super()._scale_operation(op, *args)

            lon_op, lat_op, distance_op = _spherical_op_funcs(op, *args)

            result = self.__class__(
                lon_op(self.lon),
                lat_op(self.lat),
                distance_op(self.distance),
                copy=COPY_IF_NEEDED,
            )
            for key, differential in self.differentials.items():
                new_comps = (
                    op(getattr(differential, comp))
                    for op, comp in zip(
                        (operator.pos, lat_op, distance_op), differential.components
                    )
                )
                result.differentials[key] = differential.__class__(*new_comps, copy=False)
            return result

    class PhysicsSphericalRepresentation(BaseRepresentation):
        """
        Representation of points in 3D spherical coordinates (using the physics
        convention of using ``phi`` and ``theta`` for azimuth and inclination
        from the pole).

        Parameters
        ----------
        phi, theta : `~astropy.units.Quantity` or str
            The azimuth and inclination of the point(s), in angular units. The
            inclination should be between 0 and 180 degrees, and the azimuth will
            be wrapped to an angle between 0 and 360 degrees. These can also be
            instances of `~astropy.coordinates.Angle`.  If ``copy`` is False, `phi`
            will be changed inplace if it is not between 0 and 360 degrees.

        r : `~astropy.units.Quantity`
            The distance to the point(s). If the distance is a length, it is
            passed to the :class:`~astropy.coordinates.Distance` class, otherwise
            it is passed to the :class:`~astropy.units.Quantity` class.

        differentials : dict, `~astropy.coordinates.PhysicsSphericalDifferential`, optional
            Any differential classes that should be associated with this
            representation. The input must either be a single
            `~astropy.coordinates.PhysicsSphericalDifferential` instance, or a dictionary of of
            differential instances with keys set to a string representation of the
            SI unit with which the differential (derivative) is taken. For example,
            for a velocity differential on a positional representation, the key
            would be ``'s'`` for seconds, indicating that the derivative is a time
            derivative.

        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        attr_classes = {"phi": Angle, "theta": Angle, "r": u.Quantity}

        def __init__(self, phi, theta=None, r=None, differentials=None, copy=True):
            super().__init__(phi, theta, r, copy=copy, differentials=differentials)

            # Wrap/validate phi/theta
            # Note that _phi already holds our own copy if copy=True.
            self._phi.wrap_at(360 * u.deg, inplace=True)

            if np.any(self._theta < 0.0 * u.deg) or np.any(self._theta > 180.0 * u.deg):
                raise ValueError(
                    "Inclination angle(s) must be within 0 deg <= angle <= 180 deg, "
                    f"got {theta.to(u.degree)}"
                )

            if self._r.unit.physical_type == "length":
                self._r = self._r.view(Distance)

        @property
        def phi(self):
            """
            The azimuth of the point(s).
            """
            return self._phi

        @property
        def theta(self):
            """
            The elevation of the point(s).
            """
            return self._theta

        @property
        def r(self):
            """
            The distance from the origin to the point(s).
            """
            return self._r

        def unit_vectors(self):
            sinphi, cosphi = np.sin(self.phi), np.cos(self.phi)
            sintheta, costheta = np.sin(self.theta), np.cos(self.theta)
            return {
                "phi": CartesianRepresentation(-sinphi, cosphi, 0.0, copy=COPY_IF_NEEDED),
                "theta": CartesianRepresentation(
                    costheta * cosphi, costheta * sinphi, -sintheta, copy=COPY_IF_NEEDED
                ),
                "r": CartesianRepresentation(
                    sintheta * cosphi, sintheta * sinphi, costheta, copy=COPY_IF_NEEDED
                ),
            }

        def scale_factors(self):
            r = self.r / u.radian
            sintheta = np.sin(self.theta)
            l = np.broadcast_to(1.0 * u.one, self.shape, subok=True)
            return {"phi": r * sintheta, "theta": r, "r": l}

        def represent_as(self, other_class, differential_class=None):
            # Take a short cut if the other class is a spherical representation

            if isinstance(other_class, type):
                if issubclass(other_class, SphericalRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        lon=self.phi,
                        lat=90 * u.deg - self.theta,
                        distance=self.r,
                        differentials=diffs,
                        copy=False,
                    )
                elif issubclass(other_class, UnitSphericalRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        lon=self.phi,
                        lat=90 * u.deg - self.theta,
                        differentials=diffs,
                        copy=False,
                    )
                elif issubclass(other_class, RadialRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        distance=self.r,
                        differentials=diffs,
                        copy=False,
                    )

                from .cylindrical import CylindricalRepresentation

                if issubclass(other_class, CylindricalRepresentation):
                    diffs = self._re_represent_differentials(
                        other_class, differential_class
                    )
                    return other_class(
                        rho=self.r * np.sin(self.theta),
                        phi=self.phi,
                        z=self.r * np.cos(self.theta),
                        differentials=diffs,
                        copy=False,
                    )

            return super().represent_as(other_class, differential_class)

        def to_cartesian(self):
            """
            Converts spherical polar coordinates to 3D rectangular cartesian
            coordinates.
            """
            # We need to convert Distance to Quantity to allow negative values.
            if isinstance(self.r, Distance):
                d = self.r.view(u.Quantity)
            else:
                d = self.r

            x = d * np.sin(self.theta) * np.cos(self.phi)
            y = d * np.sin(self.theta) * np.sin(self.phi)
            z = d * np.cos(self.theta)

            return CartesianRepresentation(x=x, y=y, z=z, copy=False)

        @classmethod
        def from_cartesian(cls, cart):
            """
            Converts 3D rectangular cartesian coordinates to spherical polar
            coordinates.
            """
            s = np.hypot(cart.x, cart.y)
            r = np.hypot(s, cart.z)

            phi = np.arctan2(cart.y, cart.x)
            theta = np.arctan2(s, cart.z)

            return cls(phi=phi, theta=theta, r=r, copy=False)

        def transform(self, matrix):
            """Transform the spherical coordinates using a 3x3 matrix.

            This returns a new representation and does not modify the original one.
            Any differentials attached to this representation will also be
            transformed.

            Parameters
            ----------
            matrix : (3,3) array-like
                A 3x3 matrix, such as a rotation matrix (or a stack of matrices).

            """
            # apply transformation in unit-spherical coordinates
            xyz = erfa_ufunc.s2c(self.phi, 90 * u.deg - self.theta)
            p = erfa_ufunc.rxp(matrix, xyz)
            lon, lat, ur = erfa_ufunc.p2s(p)  # `ur` is transformed unit-`r`
            # create transformed physics-spherical representation,
            # reapplying the distance scaling
            rep = self.__class__(phi=lon, theta=90 * u.deg - lat, r=self.r * ur)

            new_diffs = {
                k: d.transform(matrix, self, rep) for k, d in self.differentials.items()
            }
            return rep.with_differentials(new_diffs)

        def norm(self):
            """Vector norm.

            The norm is the standard Frobenius norm, i.e., the square root of the
            sum of the squares of all components with non-angular units.  For
            spherical coordinates, this is just the absolute value of the radius.

            Returns
            -------
            norm : `astropy.units.Quantity`
                Vector norm, with the same shape as the representation.
            """
            return np.abs(self.r)

        def _scale_operation(self, op, *args):
            if any(
                differential.base_representation is not self.__class__
                for differential in self.differentials.values()
            ):
                return super()._scale_operation(op, *args)

            phi_op, adjust_theta_sign, r_op = _spherical_op_funcs(op, *args)
            # Also run phi_op on theta to ensure theta remains between 0 and 180:
            # any time the scale is negative, we do -theta + 180 degrees.
            result = self.__class__(
                phi_op(self.phi),
                phi_op(adjust_theta_sign(self.theta)),
                r_op(self.r),
                copy=COPY_IF_NEEDED,
            )
            for key, differential in self.differentials.items():
                new_comps = (
                    op(getattr(differential, comp))
                    for op, comp in zip(
                        (operator.pos, adjust_theta_sign, r_op), differential.components
                    )
                )
                result.differentials[key] = differential.__class__(*new_comps, copy=False)
            return result

    class BaseSphericalDifferential(BaseDifferential):
        def _d_lon_coslat(self, base):
            """Convert longitude differential d_lon to d_lon_coslat.

            Parameters
            ----------
            base : instance of ``cls.base_representation``
                The base from which the latitude will be taken.
            """
            self._check_base(base)
            return self.d_lon * np.cos(base.lat)

        @classmethod
        def _get_d_lon(cls, d_lon_coslat, base):
            """Convert longitude differential d_lon_coslat to d_lon.

            Parameters
            ----------
            d_lon_coslat : `~astropy.units.Quantity`
                Longitude differential that includes ``cos(lat)``.
            base : instance of ``cls.base_representation``
                The base from which the latitude will be taken.
            """
            cls._check_base(base)
            return d_lon_coslat / np.cos(base.lat)

        def _combine_operation(self, op, other, reverse=False):
            """Combine two differentials, or a differential with a representation.

            If ``other`` is of the same differential type as ``self``, the
            components will simply be combined.  If both are different parts of
            a `~astropy.coordinates.SphericalDifferential` (e.g., a
            `~astropy.coordinates.UnitSphericalDifferential` and a
            `~astropy.coordinates.RadialDifferential`), they will combined
            appropriately.

            If ``other`` is a representation, it will be used as a base for which
            to evaluate the differential, and the result is a new representation.

            Parameters
            ----------
            op : `~operator` callable
                Operator to apply (e.g., `~operator.add`, `~operator.sub`, etc.
            other : `~astropy.coordinates.BaseRepresentation` subclass instance
                The other differential or representation.
            reverse : bool
                Whether the operands should be reversed (e.g., as we got here via
                ``self.__rsub__`` because ``self`` is a subclass of ``other``).
            """
            if (
                isinstance(other, BaseSphericalDifferential)
                and not isinstance(self, type(other))
            ) or isinstance(other, RadialDifferential):
                all_components = set(self.components) | set(other.components)
                first, second = (self, other) if not reverse else (other, self)
                result_args = {
                    c: op(getattr(first, c, 0.0), getattr(second, c, 0.0))
                    for c in all_components
                }
                return SphericalDifferential(**result_args)

            return super()._combine_operation(op, other, reverse)

    class UnitSphericalDifferential(BaseSphericalDifferential):
        """Differential(s) of points on a unit sphere.

        Parameters
        ----------
        d_lon, d_lat : `~astropy.units.Quantity`
            The longitude and latitude of the differentials.
        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        base_representation = UnitSphericalRepresentation

        @classproperty
        def _dimensional_differential(cls):
            return SphericalDifferential

        def __init__(self, d_lon, d_lat=None, copy=True):
            super().__init__(d_lon, d_lat, copy=copy)
            if not self._d_lon.unit.is_equivalent(self._d_lat.unit):
                raise u.UnitsError("d_lon and d_lat should have equivalent units.")

        @classmethod
        def from_cartesian(cls, other, base):
            # Go via the dimensional equivalent, so that the longitude and latitude
            # differentials correctly take into account the norm of the base.
            dimensional = cls._dimensional_differential.from_cartesian(other, base)
            return dimensional.represent_as(cls)

        def to_cartesian(self, base):
            if isinstance(base, SphericalRepresentation):
                scale = base.distance
            elif isinstance(base, PhysicsSphericalRepresentation):
                scale = base.r
            else:
                return super().to_cartesian(base)

            base = base.represent_as(UnitSphericalRepresentation)
            return scale * super().to_cartesian(base)

        def represent_as(self, other_class, base=None):
            # Only have enough information to represent other unit-spherical.
            if issubclass(other_class, UnitSphericalCosLatDifferential):
                return other_class(self._d_lon_coslat(base), self.d_lat)

            return super().represent_as(other_class, base)

        @classmethod
        def from_representation(cls, representation, base=None):
            # All spherical differentials can be done without going to Cartesian,
            # though CosLat needs base for the latitude.
            if isinstance(representation, SphericalDifferential):
                return cls(representation.d_lon, representation.d_lat)
            elif isinstance(
                representation,
                (SphericalCosLatDifferential, UnitSphericalCosLatDifferential),
            ):
                d_lon = cls._get_d_lon(representation.d_lon_coslat, base)
                return cls(d_lon, representation.d_lat)
            elif isinstance(representation, PhysicsSphericalDifferential):
                return cls(representation.d_phi, -representation.d_theta)

            return super().from_representation(representation, base)

        def transform(self, matrix, base, transformed_base):
            """Transform differential using a 3x3 matrix in a Cartesian basis.

            This returns a new differential and does not modify the original one.

            Parameters
            ----------
            matrix : (3,3) array-like
                A 3x3 (or stack thereof) matrix, such as a rotation matrix.
            base : instance of ``cls.base_representation``
                Base relative to which the differentials are defined.  If the other
                class is a differential representation, the base will be converted
                to its ``base_representation``.
            transformed_base : instance of ``cls.base_representation``
                Base relative to which the transformed differentials are defined.
                If the other class is a differential representation, the base will
                be converted to its ``base_representation``.
            """
            # the transformation matrix does not need to be a rotation matrix,
            # so the unit-distance is not guaranteed. For speed, we check if the
            # matrix is in O(3) and preserves lengths.
            if np.all(is_O3(matrix)):  # remain in unit-rep
                # TODO! implement without Cartesian intermediate step.
                # some of this can be moved to the parent class.
                diff = super().transform(matrix, base, transformed_base)

            else:  # switch to dimensional representation
                du = self.d_lon.unit / base.lon.unit  # derivative unit
                diff = self._dimensional_differential(
                    d_lon=self.d_lon, d_lat=self.d_lat, d_distance=0 * du
                ).transform(matrix, base, transformed_base)

            return diff

        def _scale_operation(self, op, *args, scaled_base=False):
            if scaled_base:
                return self.copy()
            else:
                return super()._scale_operation(op, *args)

    class SphericalDifferential(BaseSphericalDifferential):
        """Differential(s) of points in 3D spherical coordinates.

        Parameters
        ----------
        d_lon, d_lat : `~astropy.units.Quantity`
            The differential longitude and latitude.
        d_distance : `~astropy.units.Quantity`
            The differential distance.
        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        base_representation = SphericalRepresentation
        _unit_differential = UnitSphericalDifferential

        def __init__(self, d_lon, d_lat=None, d_distance=None, copy=True):
            super().__init__(d_lon, d_lat, d_distance, copy=copy)
            if not self._d_lon.unit.is_equivalent(self._d_lat.unit):
                raise u.UnitsError("d_lon and d_lat should have equivalent units.")

        def represent_as(self, other_class, base=None):
            # All spherical differentials can be done without going to Cartesian,
            # though CosLat needs base for the latitude.
            if issubclass(other_class, UnitSphericalDifferential):
                return other_class(self.d_lon, self.d_lat)
            elif issubclass(other_class, RadialDifferential):
                return other_class(self.d_distance)
            elif issubclass(other_class, SphericalCosLatDifferential):
                return other_class(self._d_lon_coslat(base), self.d_lat, self.d_distance)
            elif issubclass(other_class, UnitSphericalCosLatDifferential):
                return other_class(self._d_lon_coslat(base), self.d_lat)
            elif issubclass(other_class, PhysicsSphericalDifferential):
                return other_class(self.d_lon, -self.d_lat, self.d_distance)
            else:
                return super().represent_as(other_class, base)

        @classmethod
        def from_representation(cls, representation, base=None):
            # Other spherical differentials can be done without going to Cartesian,
            # though CosLat needs base for the latitude.
            if isinstance(representation, SphericalCosLatDifferential):
                d_lon = cls._get_d_lon(representation.d_lon_coslat, base)
                return cls(d_lon, representation.d_lat, representation.d_distance)
            elif isinstance(representation, PhysicsSphericalDifferential):
                return cls(
                    representation.d_phi, -representation.d_theta, representation.d_r
                )

            return super().from_representation(representation, base)

        def _scale_operation(self, op, *args, scaled_base=False):
            if scaled_base:
                return self.__class__(self.d_lon, self.d_lat, op(self.d_distance, *args))
            else:
                return super()._scale_operation(op, *args)

    class BaseSphericalCosLatDifferential(BaseDifferential):
        """Differentials from points on a spherical base representation.

        With cos(lat) assumed to be included in the longitude differential.
        """

        @classmethod
        def _get_base_vectors(cls, base):
            """Get unit vectors and scale factors from (unit)spherical base.

            Parameters
            ----------
            base : instance of ``self.base_representation``
                The points for which the unit vectors and scale factors should be
                retrieved.

            Returns
            -------
            unit_vectors : dict of `~astropy.coordinates.CartesianRepresentation`
                In the directions of the coordinates of base.
            scale_factors : dict of `~astropy.units.Quantity`
                Scale factors for each of the coordinates.  The scale factor for
                longitude does not include the cos(lat) factor.

            Raises
            ------
            TypeError : if the base is not of the correct type
            """
            cls._check_base(base)
            return base.unit_vectors(), base.scale_factors(omit_coslat=True)

        def _d_lon(self, base):
            """Convert longitude differential with cos(lat) to one without.

            Parameters
            ----------
            base : instance of ``cls.base_representation``
                The base from which the latitude will be taken.
            """
            self._check_base(base)
            return self.d_lon_coslat / np.cos(base.lat)

        @classmethod
        def _get_d_lon_coslat(cls, d_lon, base):
            """Convert longitude differential d_lon to d_lon_coslat.

            Parameters
            ----------
            d_lon : `~astropy.units.Quantity`
                Value of the longitude differential without ``cos(lat)``.
            base : instance of ``cls.base_representation``
                The base from which the latitude will be taken.
            """
            cls._check_base(base)
            return d_lon * np.cos(base.lat)

        def _combine_operation(self, op, other, reverse=False):
            """Combine two differentials, or a differential with a representation.

            If ``other`` is of the same differential type as ``self``, the
            components will simply be combined.  If both are different parts of
            a `~astropy.coordinates.SphericalDifferential` (e.g., a
            `~astropy.coordinates.UnitSphericalDifferential` and a
            `~astropy.coordinates.RadialDifferential`), they will combined
            appropriately.

            If ``other`` is a representation, it will be used as a base for which
            to evaluate the differential, and the result is a new representation.

            Parameters
            ----------
            op : `~operator` callable
                Operator to apply (e.g., `~operator.add`, `~operator.sub`, etc.
            other : `~astropy.coordinates.BaseRepresentation` subclass instance
                The other differential or representation.
            reverse : bool
                Whether the operands should be reversed (e.g., as we got here via
                ``self.__rsub__`` because ``self`` is a subclass of ``other``).
            """
            if (
                isinstance(other, BaseSphericalCosLatDifferential)
                and not isinstance(self, type(other))
            ) or isinstance(other, RadialDifferential):
                all_components = set(self.components) | set(other.components)
                first, second = (self, other) if not reverse else (other, self)
                result_args = {
                    c: op(getattr(first, c, 0.0), getattr(second, c, 0.0))
                    for c in all_components
                }
                return SphericalCosLatDifferential(**result_args)

            return super()._combine_operation(op, other, reverse)

    class UnitSphericalCosLatDifferential(BaseSphericalCosLatDifferential):
        """Differential(s) of points on a unit sphere.

        Parameters
        ----------
        d_lon_coslat, d_lat : `~astropy.units.Quantity`
            The longitude and latitude of the differentials.
        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        base_representation = UnitSphericalRepresentation
        attr_classes = {"d_lon_coslat": u.Quantity, "d_lat": u.Quantity}

        @classproperty
        def _dimensional_differential(cls):
            return SphericalCosLatDifferential

        def __init__(self, d_lon_coslat, d_lat=None, copy=True):
            super().__init__(d_lon_coslat, d_lat, copy=copy)
            if not self._d_lon_coslat.unit.is_equivalent(self._d_lat.unit):
                raise u.UnitsError("d_lon_coslat and d_lat should have equivalent units.")

        @classmethod
        def from_cartesian(cls, other, base):
            # Go via the dimensional equivalent, so that the longitude and latitude
            # differentials correctly take into account the norm of the base.
            dimensional = cls._dimensional_differential.from_cartesian(other, base)
            return dimensional.represent_as(cls)

        def to_cartesian(self, base):
            if isinstance(base, SphericalRepresentation):
                scale = base.distance
            elif isinstance(base, PhysicsSphericalRepresentation):
                scale = base.r
            else:
                return super().to_cartesian(base)

            base = base.represent_as(UnitSphericalRepresentation)
            return scale * super().to_cartesian(base)

        def represent_as(self, other_class, base=None):
            # Only have enough information to represent other unit-spherical.
            if issubclass(other_class, UnitSphericalDifferential):
                return other_class(self._d_lon(base), self.d_lat)

            return super().represent_as(other_class, base)

        @classmethod
        def from_representation(cls, representation, base=None):
            # All spherical differentials can be done without going to Cartesian,
            # though w/o CosLat needs base for the latitude.
            if isinstance(representation, SphericalCosLatDifferential):
                return cls(representation.d_lon_coslat, representation.d_lat)
            elif isinstance(
                representation, (SphericalDifferential, UnitSphericalDifferential)
            ):
                d_lon_coslat = cls._get_d_lon_coslat(representation.d_lon, base)
                return cls(d_lon_coslat, representation.d_lat)
            elif isinstance(representation, PhysicsSphericalDifferential):
                d_lon_coslat = cls._get_d_lon_coslat(representation.d_phi, base)
                return cls(d_lon_coslat, -representation.d_theta)

            return super().from_representation(representation, base)

        def transform(self, matrix, base, transformed_base):
            """Transform differential using a 3x3 matrix in a Cartesian basis.

            This returns a new differential and does not modify the original one.

            Parameters
            ----------
            matrix : (3,3) array-like
                A 3x3 (or stack thereof) matrix, such as a rotation matrix.
            base : instance of ``cls.base_representation``
                Base relative to which the differentials are defined.  If the other
                class is a differential representation, the base will be converted
                to its ``base_representation``.
            transformed_base : instance of ``cls.base_representation``
                Base relative to which the transformed differentials are defined.
                If the other class is a differential representation, the base will
                be converted to its ``base_representation``.
            """
            # the transformation matrix does not need to be a rotation matrix,
            # so the unit-distance is not guaranteed. For speed, we check if the
            # matrix is in O(3) and preserves lengths.
            if np.all(is_O3(matrix)):  # remain in unit-rep
                # TODO! implement without Cartesian intermediate step.
                diff = super().transform(matrix, base, transformed_base)

            else:  # switch to dimensional representation
                du = self.d_lat.unit / base.lat.unit  # derivative unit
                diff = self._dimensional_differential(
                    d_lon_coslat=self.d_lon_coslat, d_lat=self.d_lat, d_distance=0 * du
                ).transform(matrix, base, transformed_base)

            return diff

        def _scale_operation(self, op, *args, scaled_base=False):
            if scaled_base:
                return self.copy()
            else:
                return super()._scale_operation(op, *args)

    class SphericalCosLatDifferential(BaseSphericalCosLatDifferential):
        """Differential(s) of points in 3D spherical coordinates.

        Parameters
        ----------
        d_lon_coslat, d_lat : `~astropy.units.Quantity`
            The differential longitude (with cos(lat) included) and latitude.
        d_distance : `~astropy.units.Quantity`
            The differential distance.
        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        base_representation = SphericalRepresentation
        _unit_differential = UnitSphericalCosLatDifferential
        attr_classes = {
            "d_lon_coslat": u.Quantity,
            "d_lat": u.Quantity,
            "d_distance": u.Quantity,
        }

        def __init__(self, d_lon_coslat, d_lat=None, d_distance=None, copy=True):
            super().__init__(d_lon_coslat, d_lat, d_distance, copy=copy)
            if not self._d_lon_coslat.unit.is_equivalent(self._d_lat.unit):
                raise u.UnitsError("d_lon_coslat and d_lat should have equivalent units.")

        def represent_as(self, other_class, base=None):
            # All spherical differentials can be done without going to Cartesian,
            # though some need base for the latitude to remove cos(lat).
            if issubclass(other_class, UnitSphericalCosLatDifferential):
                return other_class(self.d_lon_coslat, self.d_lat)
            elif issubclass(other_class, RadialDifferential):
                return other_class(self.d_distance)
            elif issubclass(other_class, SphericalDifferential):
                return other_class(self._d_lon(base), self.d_lat, self.d_distance)
            elif issubclass(other_class, UnitSphericalDifferential):
                return other_class(self._d_lon(base), self.d_lat)
            elif issubclass(other_class, PhysicsSphericalDifferential):
                return other_class(self._d_lon(base), -self.d_lat, self.d_distance)

            return super().represent_as(other_class, base)

        @classmethod
        def from_representation(cls, representation, base=None):
            # Other spherical differentials can be done without going to Cartesian,
            # though we need base for the latitude to remove coslat.
            if isinstance(representation, SphericalDifferential):
                d_lon_coslat = cls._get_d_lon_coslat(representation.d_lon, base)
                return cls(d_lon_coslat, representation.d_lat, representation.d_distance)
            elif isinstance(representation, PhysicsSphericalDifferential):
                d_lon_coslat = cls._get_d_lon_coslat(representation.d_phi, base)
                return cls(d_lon_coslat, -representation.d_theta, representation.d_r)

            return super().from_representation(representation, base)

        def _scale_operation(self, op, *args, scaled_base=False):
            if scaled_base:
                return self.__class__(
                    self.d_lon_coslat, self.d_lat, op(self.d_distance, *args)
                )
            else:
                return super()._scale_operation(op, *args)

    class RadialDifferential(BaseDifferential):
        """Differential(s) of radial distances.

        Parameters
        ----------
        d_distance : `~astropy.units.Quantity`
            The differential distance.
        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        base_representation = RadialRepresentation

        def to_cartesian(self, base):
            unit_vec = base.represent_as(UnitSphericalRepresentation).to_cartesian()
            return self.d_distance * unit_vec

        def norm(self, base=None):
            return self.d_distance

        @classmethod
        def from_cartesian(cls, other, base):
            return cls(
                other.dot(base.represent_as(UnitSphericalRepresentation)), copy=False
            )

        @classmethod
        def from_representation(cls, representation, base=None):
            if isinstance(
                representation, (SphericalDifferential, SphericalCosLatDifferential)
            ):
                return cls(representation.d_distance)
            elif isinstance(representation, PhysicsSphericalDifferential):
                return cls(representation.d_r)
            else:
                return super().from_representation(representation, base)

        def _combine_operation(self, op, other, reverse=False):
            if isinstance(other, self.base_representation):
                if reverse:
                    first, second = other.distance, self.d_distance
                else:
                    first, second = self.d_distance, other.distance
                return other.__class__(op(first, second), copy=False)
            elif isinstance(
                other, (BaseSphericalDifferential, BaseSphericalCosLatDifferential)
            ):
                all_components = set(self.components) | set(other.components)
                first, second = (self, other) if not reverse else (other, self)
                result_args = {
                    c: op(getattr(first, c, 0.0), getattr(second, c, 0.0))
                    for c in all_components
                }
                return SphericalDifferential(**result_args)

            else:
                return super()._combine_operation(op, other, reverse)

    class PhysicsSphericalDifferential(BaseDifferential):
        """Differential(s) of 3D spherical coordinates using physics convention.

        Parameters
        ----------
        d_phi, d_theta : `~astropy.units.Quantity`
            The differential azimuth and inclination.
        d_r : `~astropy.units.Quantity`
            The differential radial distance.
        copy : bool, optional
            If `True` (default), arrays will be copied. If `False`, arrays will
            be references, though possibly broadcast to ensure matching shapes.
        """

        base_representation = PhysicsSphericalRepresentation

        def __init__(self, d_phi, d_theta=None, d_r=None, copy=True):
            super().__init__(d_phi, d_theta, d_r, copy=copy)
            if not self._d_phi.unit.is_equivalent(self._d_theta.unit):
                raise u.UnitsError("d_phi and d_theta should have equivalent units.")

        def represent_as(self, other_class, base=None):
            # All spherical differentials can be done without going to Cartesian,
            # though CosLat needs base for the latitude. For those, explicitly
            # do the equivalent of self._d_lon_coslat in SphericalDifferential.
            if issubclass(other_class, SphericalDifferential):
                return other_class(self.d_phi, -self.d_theta, self.d_r)
            elif issubclass(other_class, UnitSphericalDifferential):
                return other_class(self.d_phi, -self.d_theta)
            elif issubclass(other_class, SphericalCosLatDifferential):
                self._check_base(base)
                d_lon_coslat = self.d_phi * np.sin(base.theta)
                return other_class(d_lon_coslat, -self.d_theta, self.d_r)
            elif issubclass(other_class, UnitSphericalCosLatDifferential):
                self._check_base(base)
                d_lon_coslat = self.d_phi * np.sin(base.theta)
                return other_class(d_lon_coslat, -self.d_theta)
            elif issubclass(other_class, RadialDifferential):
                return other_class(self.d_r)

            return super().represent_as(other_class, base)

        @classmethod
        def from_representation(cls, representation, base=None):
            # Other spherical differentials can be done without going to Cartesian,
            # though we need base for the latitude to remove coslat. For that case,
            # do the equivalent of cls._d_lon in SphericalDifferential.
            if isinstance(representation, SphericalDifferential):
                return cls(
                    representation.d_lon, -representation.d_lat, representation.d_distance
                )
            elif isinstance(representation, SphericalCosLatDifferential):
                cls._check_base(base)
                d_phi = representation.d_lon_coslat / np.sin(base.theta)
                return cls(d_phi, -representation.d_lat, representation.d_distance)

            return super().from_representation(representation, base)

        def _scale_operation(self, op, *args, scaled_base=False):
            if scaled_base:
                return self.__class__(self.d_phi, self.d_theta, op(self.d_r, *args))
            else:
                return super()._scale_operation(op, *args)