# Source code for spacepy.coordinates

    #!/usr/bin/env python
    # -*- coding: utf-8 -*-
    """Implementation of Coords class functions for coordinate transformations

    The coordinate systems supported by this module cover the most commonly
    used geophysical and magnetospheric systems. The naming conventions can
    follow the names used by the popular IRBEM library, but for inertial
    systems we use a more consistent, fine-grained naming convention that
    clarifies the different systems.

        Earth-centered Inertial Systems
        -------------------------------
        * **ECI2000** Earth-centered Inertial, J2000 epoch
        * **ECIMOD** Earth-centered Inertial, mean-of-date
        * **ECITOD** Earth-centered Inertial, true-of-date
        * **GEI** Geocentric Equatorial Inertial (IRBEM approximation of TOD)

        Magnetospheric Systems
        ----------------------
        * **GSM** Geocentric Solar Magnetospheric
        * **GSE** Geocentric Solar Ecliptic
        * **SM** Solar Magnetic
        * **MAG** Geomagnetic Coordinate System (aka CDMAG)

        Earth-fixed Systems
        -------------------
        * **GEO** Geocentric geographic, aka Earth-centered Earth-fixed
        * **GDZ** Geodetic coordinates

    By convention *all* systems are treated as natively Cartesian except
    geodetic (GDZ), which is defined in [altitude, latitude, longitude]
    where altitude is relative to a reference ellipsoid. Similarly, distance
    units are assumed to be Earth radii (Re) in all systems except GDZ, where
    altitude is given in km. Conversions to GDZ will output altitude in km
    regardless of the input distance units and conversions from GDZ will
    output in Re regardless of input units. In all other cases, the distance
    units will be preserved.

        .. versionchanged:: 0.3.0

        The new CTrans backend was added, which includes support for the names
        ``ECI2000``, ``ECIMOD``, ``ECITOD``, and ``CDMAG``. With the exception
        of ``ECIMOD``, these can be used with the existing IRBEM backend, and
        will be converted to their closest equivalents.

        .. versionchanged:: 0.4.0

        The default backend for coordinate transformations was changed from IRBEM
        to the CTrans-based SpacePy backend.

    Notes on differences between representations
    --------------------------------------------
    IRBEM's coordinate transformations are low-accuracy and were written for
    a library with a driving philosophy of speed and robustness as priorities.
    The coordinate transformations are therefore approximate. Further, most of
    the geophysical systems (e.g., GSE, SM) are derived from an inertial
    system. It is standard practice to use ECIMOD as this system. However,
    IRBEM does not currently make ECIMOD available as one of its inertial
    systems. IRBEM's default inertial system (called GEI) is consistent with
    an approximation of ECITOD. Hence there will be small differences between
    IRBEM's transformations and those using SpacePy's CTrans backend.
    Further sources of difference include: IRBEM uses a low-order approximation
    to the sidereal time and other parameters; the calculation of the Earth-Sun
    vector differs between the representations; the definitions of an Earth
    radius differ (SpacePy = 6378.137km; IRBEM = 6371.2 km). SpacePy's in-built
    representation is higher accuracy and is comprehensively tested, including
    tests for consistency with other high accuracy packages such as LANLGeoMag
    and AstroPy. However, for use cases where the required precision is of order
    1 percent the output can be considered equivalent.

    Setting options for coordinate transformation
    ---------------------------------------------
    The backend for coordinate transformations can be provided at
    instantiation of a :class:`Coords` object using a keyword
    argument. However, for convenience and flexibility the options can be
    set at the module level. Configurable options include the backend used
    (:mod:`~spacepy.irbempy` or SpacePy's :mod:`~spacepy.ctrans`) and the
    reference ellipsoid (only configurable for the SpacePy backend). A
    warning will be raised if the backend is not set (either through the
    defaults or the keyword argument). The final configurable option
    (``itol``) is the maximum separation, in seconds, for which the
    coordinate transformations will not be recalculated. To force all
    transformations to use an exact transform for the time, set ``itol``
    to zero. Values between 10s and 60s are recommended for speed while
    also preserving accuracy, though different applications will require
    different accuracies.  For example, assuming this module has been
    imported as ``spc``, to set the SpacePy backend as the default and set
    ``itol`` to 5 seconds:

        >>> spc.DEFAULTS.set_values(use_irbem=False, itol=5)

    Authors: Steven Morley and Josef Koller
    Institution: Los ALamos National Laboratory
    Contact: smorley@lanl.gov

    Copyright 2010-2016 Los Alamos National Security, LLC.

    """

    from collections import namedtuple
    import numbers
    import warnings
    import numpy as np
    from spacepy import help
    import spacepy
    import spacepy.datamodel as dm
    import spacepy.time
    from spacepy import ctrans

    __contact__ = 'Steven Morley, smorley@lanl.gov'

    # -----------------------------------------------
    # space coordinate class
    # -----------------------------------------------

    SYSAXES_TYPES = {'GDZ': {'sph': 0, 'car': None},
                     'GEO': {'sph': None, 'car': 1}, 'GSM': {'sph': None, 'car': 2},
                     'GSE': {'sph': None, 'car': 3}, 'SM': {'sph': None, 'car': 4},
                     'GEI': {'sph': None, 'car': 5}, 'CDMAG': {'sph': None, 'car': 6},
                     'MAG': {'sph': None, 'car': 6}, 'SPH': {'sph': 7, 'car': None},
                     'RLL': {'sph': 8, 'car': None}, 'TOD': {'sph': None, 'car': 12},
                     'ECITOD': {'sph': None, 'car': 12}, 'J2000': {'sph': None, 'car': 13},
                     'ECI2000': {'sph': None, 'car': 13}, 'TEME': {'sph': None, 'car': 14},
                     'ECIMOD': {'sph': None, 'car': 15}}

    SYS_EQUIV = {'SPH': 'GEO', 'GEI': 'ECITOD', 'TOD': 'ECITOD',
                 'J2000': 'ECI2000', 'MAG': 'CDMAG'}

    IRBEM_RE = 6371.2  # kilometers

    class __Defaults(object):
        '''Configuration factory for coordinates module default settings
        '''
        def __init__(self):
            self._fac = namedtuple('values', 'use_irbem, ellipsoid, itol')
            self.set_values()

        def __repr__(self):
            full = '{}'.format(self.values)
            b1 = full.index('(')
            return 'DEFAULTS{}'.format(full[b1:])

        def set_values(self, use_irbem=None, ellipsoid=ctrans.WGS84, itol=30):
            '''Set options for coordinate transforms

            Parameters
            ----------
            use_irbem : bool
                If True, use IRBEM as coordinate transform backend. If False, use SpacePy's
                implementations. Default is to use SpacePy.
            ellipsoid : spacepy.ctrans.Ellipsoid
                A reference ellipsoid to use for the Earth radius definition and for geodetic
                coordinate conversion. SpacePy defaults to the WGS84 ellipsoid and the semi-major
                axis as the Earth radius (6738.137 km). The IRBEM backend uses WGS84, but expresses
                an Earth radius using 6371.2 km.
            '''
            self.values = self._fac(use_irbem=use_irbem,
                                    ellipsoid=ellipsoid,
                                    itol=itol)

    DEFAULTS = __Defaults()

    class Coords(object):
        '''
        a = Coords( data, dtype, carsph, [units, ticks, use_irbem])

        A class holding spatial coordinates and enabling transformation between
        coordinate systems. Coordinates can be stored as Cartesian or spherical
        and units are assumed to be Re (distance) and degrees (angle)

        .. note:: Although other units may be specified and will be carried
             through, most functions throughout SpacePy assume distances
             in Re and angles in degrees, regardless of specified units.

        By default, coordinate transforms are based on the SpacePy library's
        high-accuracy coordinates backend.
        The legacy transforms provided by the IRBEM library can also be used
        by setting the `use_irbem` flag to True; its manual
        <http://svn.code.sf.net/p/irbem/code/trunk/manual/user_guide.html>`_
        may prove useful.
        For a good reference on heliospheric and magnetospheric
        coordinate systems, see Franz & Harper, "Heliospheric Coordinate Systems",
        Planet. Space Sci., 50, pp 217-233, 2002
        (https://doi.org/10.1016/S0032-0633(01)00119-2).

        Parameters
        ----------
        data : list or ndarray, dim = (n,3)
            coordinate points [X,Y,Z] or [rad, lat, lon]
        dtype : string
            coordinate system; supported systems are defined in
            module-level documentation. Common systems include
            GEO, GSE, GSM, SM, MAG, ECIMOD
        carsph : string
            Cartesian or spherical, 'car' or 'sph'
        units : list of strings, optional
            standard are  ['Re', 'Re', 'Re'] or ['Re', 'deg', 'deg'] depending
            on the carsph content. See note.
        ticks : Ticktock instance, optional
            used for coordinate transformations (see a.convert)
        use_irbem : bool

            .. versionadded:: 0.3.0

            Set to True to use IRBEM for coordinate transforms.
            Otherwise use SpacePy's coordinate transform library.

        Returns
        -------
        out : Coords instance
            instance with a.data, a.carsph, etc.

        See Also
        --------
        spacepy.coordinates.DEFAULTS
        spacepy.time.Ticktock

        Examples
        --------
        >>> from spacepy import coordinates as coord
        >>> cvals = coord.Coords([[1,2,4],[1,2,2]], 'GEO', 'car')
        >>> cvals.x  # returns all x coordinates
        array([1, 1])
        >>> from spacepy.time import Ticktock
        >>> cvals.ticks = Ticktock(['2002-02-02T12:00:00', '2002-02-02T12:00:00'], 'ISO') # add ticks
        >>> newcoord = cvals.convert('GSM', 'sph')
        >>> newcoord

        .. currentmodule:: spacepy.coordinates
        .. autosummary::
            ~Coords.append
            ~Coords.convert
            ~Coords.from_skycoord
            ~Coord.to_skycoord
        .. automethod:: append
        .. automethod:: convert
        .. automethod:: from_skycoord
        .. automethod:: to_skycoord
        '''
        def __init__(self, data, dtype, carsph, units=None, ticks=None, use_irbem=None):
            if use_irbem is None:
                use_irbem = DEFAULTS.values.use_irbem
            if use_irbem is None:
                use_irbem = False
                warnings.warn('No coordinate backend specified; using SpacePy.'
                              ' This default changed from IRBEM in version 0.4.0',
                              DeprecationWarning)
            if use_irbem:
                from . import irbempy as op
            self.use_irbem = use_irbem
            # setup units
            self.Re = IRBEM_RE if use_irbem else DEFAULTS.values.ellipsoid['A']  # kilometers

            # Make sure that inputs are all formed correctly
            self.data = np.atleast_2d(data).astype(np.float_)
            if len(self.data.shape) != 2 or self.data.shape[-1] != 3:
                raise ValueError('Input position vectors must be Nx3')

            dtype = dtype.upper()
            carsph = carsph.lower()

            if dtype not in list(SYSAXES_TYPES.keys()):
                raise ValueError('This dtype=' + dtype + ' is not supported. Only ' +
                                 str(list(SYSAXES_TYPES.keys()))
                                 )
            if carsph not in ['car', 'sph']:
                raise ValueError('This carsph=' + str(carsph) +
                                 ' is not supported. Only "car" or "sph"')

            if dtype in ['GDZ', 'RLL'] and carsph == 'car':
                raise ValueError("Geodetic coordinates are referenced to an ellipsoid "
                                 + "and can not be properly represented as Cartesian")

            # add ticks
            if ticks:
                assert len(ticks) == len(self.data), 'Ticktock dimensions seem off'
            self.ticks = ticks

            # GEO and SPH are cartesian/spherical versions of GEO, so use GEO for everything
            if dtype == 'SPH':
                dtype = 'GEO'
            # Make sure we're using the SpacePy name if we're using the SpacePy backend
            self.dtype = SYS_EQUIV[dtype] if (not use_irbem and dtype in SYS_EQUIV) else dtype
            self.sysaxes = SYSAXES_TYPES[dtype][carsph]

            self.carsph = carsph
            # setup units
            if units is None and carsph == 'car':
                # use standard units
                self.units = ['Re', 'Re', 'Re']
            elif units is None and carsph == 'sph':
                self.units = ['Re', 'deg', 'deg']
            else:
                self.units = units
            if dtype == 'GDZ' and carsph == 'sph':
                self.units = ['km', 'deg', 'deg']
            # setup x,y,z etc
            if carsph == 'car':
                self.x = spacepy.datamodel.dmcopy(self.data[:, 0])
                self.y = spacepy.datamodel.dmcopy(self.data[:, 1])
                self.z = spacepy.datamodel.dmcopy(self.data[:, 2])
            else:
                self.radi = spacepy.datamodel.dmcopy(self.data[:, 0])
                self.lati = spacepy.datamodel.dmcopy(self.data[:, 1])
                self.long = spacepy.datamodel.dmcopy(self.data[:, 2])
            self.shape = np.shape(self.data)
            return None

        # -----------------------------------------------
        def __str__(self):
            '''
            a.__str__() or a

            Will be called when printing Coords instance a

            Returns
            -------
            out : string
                string represenation of the instance

            Examples
            --------
            >>> from spacepy.coordinates import Coords
            >>> y = Coords([[1,2,4],[1,2,2]], 'GEO', 'car')
            >>> y
            Coords( [[1 2 4]
             [1 2 2]] ), dtype=GEO,car, units=['Re', 'Re', 'Re']
            '''
            rstr = "Coords( {0} , '{1}', '{2}')".format(self.data.tolist(), self.dtype, self.carsph)
            return rstr
        __repr__ = __str__

        # -----------------------------------------------
        def __getitem__(self, idx):
            '''
            a.__getitem__(idx) or a[idx]

            Will be called when requesting items in this instance

            Parameters
            ----------
            idx : int
                integer numbers as index

            Returns
            -------
            out : numpy array
                new values

            Examples
            --------
            >>> from spacepy.coordinates import Coords
            >>> y = Coords([[1,2,4],[1,2,2]], 'GEO', 'car')
            >>> y[0]
            Coords( [[1 2 4]] ), dtype=GEO,car, units=['Re', 'Re', 'Re']
            '''
            arr = np.array(self.data)
            t_select = self.ticks[idx] if self.ticks else self.ticks

            return Coords(arr[idx].tolist(), self.dtype, self.carsph, self.units,
                          t_select, use_irbem=self.use_irbem)

        # -----------------------------------------------
        def __setitem__(self, idx, vals):
            '''
            a.__setitem__(idx, vals) or a[idx] = vals

            Will be called setting items in this instance

            Parameters
            ----------
            idx : int
                integer numbers as index
            vals : numpy array or list
                new values

            Examples
            --------
            >>> from spacepy.coordinates import Coords
            >>> y = Coords([[1,2,4],[1,2,2]], 'GEO', 'car')
            >>> y[1] = [9,9,9]
            >>> y
            Coords( [[1 2 4]
                [9 9 9]] ), dtype=GEO,car, units=['Re', 'Re', 'Re']
            '''
            self.data[idx] = vals
            return

        # -----------------------------------------------
        def __len__(self):
            '''
            a.__len__() or len(a)

            Will be called when requesting the length, i.e. number of items

            Returns
            -------
            out : int
                length

            Examples
            --------
            >>> from spacepy.coordinates import Coords
            >>> y = Coords([[1,2,4],[1,2,2]], 'GEO', 'car')
            >>> len(y)
            2

            '''
            if isinstance(self.data, (list, np.ndarray)):
                return len(self.data)
            else:
                return 1
            return

        # -----------------------------------------------
        def convert(self, returntype, returncarsph):
            '''Create a new Coords instance with new coordinate types

            Parameters
            ----------
            returntype : string
                coordinate system, see module level documentation for
                supported systems
            returncarsph : string
                coordinate type, possible 'car' for Cartesian and 'sph' for spherical

            Returns
            -------
            out : Coords object
                Coords object in the new coordinate system

            Examples
            --------
            >>> from spacepy.coordinates import Coords
            >>> y = Coords([[1,2,4],[1,2,2]], 'GEO', 'car')
            >>> from spacepy.time import Ticktock
            >>> y.ticks = Ticktock(['2002-02-02T12:00:00', '2002-02-02T12:00:00'], 'ISO')
            >>> x = y.convert('SM','car')
            >>> x
            Coords( [[ 0.81134097  2.6493305   3.6500375 ]
             [ 0.92060408  2.30678864  1.68262126]] ), dtype=SM,car, units=['Re', 'Re', 'Re']
            '''
            # Check whether the name is an equivalent IRBEM name
            returnname = SYS_EQUIV[returntype] if returntype in SYS_EQUIV else returntype
            is_dtype_returntype = (self.dtype == returntype) or (self.dtype == returnname)

            if returntype in ['GDZ', 'RLL'] and returncarsph == 'car':
                raise ValueError("Geodetic coordinates are referenced to an ellipsoid "
                                 + "and can not be properly represented as Cartesian")

            # Then check whether a full coordinate transformation is required
            if is_dtype_returntype:
                if self.carsph == returncarsph:
                    # no change necessary
                    return self

                if self.carsph != returncarsph:
                    # only car2sph or sph2car is needed
                    if returncarsph == 'car':
                        carsph = 'car'
                        units = [self.units[0]]*3
                        data = sph2car(self.data)
                    else:
                        carsph = 'sph'
                        units = [self.units[0], 'deg', 'deg']
                        data = car2sph(self.data)
                    return Coords(data, self.dtype, carsph, units, self.ticks,
                                  use_irbem=self.use_irbem)

            # check the length of ticks and do the more complex conversions
            if self.ticks and (len(self.ticks) != len(self)):
                raise ValueError('Ticktock dimension does not match Coords dimensions')

            # check if car2sph is needed first for oneralib compatibility
            if (self.sysaxes is None):  # a car2sph or sph2car is needed
                if self.carsph == 'sph':
                    carsph = 'car'
                    units = [self.units[0]]*3
                    data = sph2car(self.data)
                else:
                    carsph = 'sph'
                    units = [self.units[0], 'deg', 'deg']
                    data = car2sph(self.data)
            else:
                data = self.data.copy()
                units = dm.dmcopy(self.units)
                carsph = dm.dmcopy(self.carsph)

            if self.use_irbem:
                from . import irbempy as op
                # irbempy.coord_trans needs the passed Coords object to have the "from" dtype
                NewCoords = Coords(data, self.dtype, carsph, units, self.ticks,
                                   use_irbem=self.use_irbem)
            else:
                # SpacePy conversion takes input/output type separately, so set to desired output
                NewCoords = Coords(data, returntype if self.use_irbem else returnname,
                                   returncarsph, units, self.ticks, use_irbem=self.use_irbem)

            # now convert to other coordinate system
            if not is_dtype_returntype:
                assert self.ticks, "Time information required; add a .ticks attribute"
                if self.use_irbem:
                    if returnname in ['GDZ', 'RLL'] and units[0] == 'km':
                        NewCoords.data /= self.Re  # IRBEM geodetic calculation requires Re input
                    NewCoords.data = op.coord_trans(NewCoords, returntype, returncarsph)
                    NewCoords.dtype = returntype
                    if returnname == 'RLL' and units[0] == 'km':
                        NewCoords.data[:, 0] *= self.Re  # Preserve units
                else:
                    if returnname in ['GDZ', 'RLL'] and self.units[0] == 'Re':
                        # converting from a Cartesian referenced system to an
                        # ellipsoid referenced system, input must be in km
                        data *= self.Re  # geodetic is defined in [km, deg, deg]
                    NewCoords.data = np.atleast_2d(ctrans.convert_multitime(data, self.ticks,
                                                                            self.dtype, returnname,
                                                                            DEFAULTS.values))
                    if returnname == 'RLL' and units[0] == 'Re':
                        # RLL is defined in Re, deg, deg, but all geodetic calcs are in km
                        NewCoords.data[:, 0] = NewCoords.data[:, 0]/self.Re
                    if (self.dtype in ['RLL', 'GDZ'] and
                        returnname not in ['RLL', 'GDZ'] and
                        units[0] == 'km'):
                        # input was in km, needs to be Earth radii
                        NewCoords.data /= self.Re
                    if returncarsph == 'sph' and returnname not in ['GDZ', 'RLL']:
                        NewCoords.data = car2sph(NewCoords.data)
                if self.dtype == 'GDZ':  # GDZ is defined in km and everything else is strictly Re
                    units[0] = 'Re'  # Converting back from GDZ set units to Re
                NewCoords.carsph = returncarsph
                NewCoords.sysaxes = SYSAXES_TYPES[NewCoords.dtype][NewCoords.carsph]

            # Correct corresponding attributes and units
            if returncarsph == 'sph':
                if returnname == 'GDZ':
                    NewCoords.units = ['km', 'deg', 'deg']
                else:
                    NewCoords.units = [units[0], 'deg', 'deg']
                for k in ('x', 'y', 'z'):
                    if hasattr(NewCoords, k):
                        delattr(NewCoords, k)
                NewCoords.radi = NewCoords.data[:, 0]
                NewCoords.lati = NewCoords.data[:, 1]
                NewCoords.long = NewCoords.data[:, 2]
            else:  # 'car'
                NewCoords.units = [units[0]]*3
                for k in ('radi', 'lati', 'long'):
                    if hasattr(NewCoords, k):
                        delattr(NewCoords, k)
                NewCoords.x = NewCoords.data[:, 0]
                NewCoords.y = NewCoords.data[:, 1]
                NewCoords.z = NewCoords.data[:, 2]

            return NewCoords

        # -----------------------------------------------
        def __getstate__(self):
            '''
            Is called when pickling
            See Also http://docs.python.org/library/pickle.html
            '''
            odict = self.__dict__.copy()  # copy the dict since we change it
            return odict

        def __setstate__(self, dict):
            '''
            Is called when unpickling
            See Also http://docs.python.org/library/pickle.html
            '''
            self.__dict__.update(dict)
            return

        # -----------------------------------------------
        def append(self, other):
            '''Append another Coords instance to the current one

            Parameters
            ----------
            other : Coords instance
                Coords instance to append

            Examples
            --------
            '''
            data = list(self.data)
            otherdata = other.convert(self.dtype, self.carsph)
            data.extend(list(otherdata.data))
            with warnings.catch_warnings():
                warnings.filterwarnings('ignore', message=r'Use of IRBEM to perform',
                                        category=DeprecationWarning,
                                        module=r'spacepy.coordinates$')
                newobj = Coords(data, dtype=self.dtype, carsph=self.carsph, use_irbem=self.use_irbem)
            return newobj

        # -----------------------------------------------
        def to_skycoord(self):
            '''
            Create an Astropy SkyCoord instance based on this instance

            Returns
            =======
            out : `astropy.coordinates.SkyCoord`
                This coordinate as an Astropy SkyCoord

            Notes
            =====
            This method requires Astropy to be installed.

            This method uses the GEO coordinate frame as the common frame between the two libraries.
            '''
            if self.ticks is None:
                raise ValueError("This method requires the attribute `ticks` to be specified.")

            try:
                import astropy.coordinates
                import astropy.time
            except ImportError as e:
                raise e.__class__("This method requires Astropy to be installed.")

            coord = self.convert('GEO', 'car')  # GEO is Astropy ITRS
            data = coord.data * self.Re * 1000 # convert to meters
            obstime = coord.ticks.APT

            return astropy.coordinates.SkyCoord(x=data[:, 0], y=data[:, 1], z=data[:, 2],
                                                unit='m', frame='itrs', obstime=obstime)

        # -----------------------------------------------
        @classmethod
        def from_skycoord(cls, skycoord, use_irbem=None):
            '''
            Create a Coords instance from an Astropy SkyCoord instance

            Parameters
            ==========
            skycoord : `astropy.coordinates.SkyCoord`
                The coordinate to be converted

            Returns
            =======
            out : Coords instance
                The converted coordinate

            Notes
            =====
            This method requires Astropy to be installed.

            This method uses the GEO coordinate frame as the common frame between the two libraries.
            '''
            try:
                import astropy.coordinates
            except ImportError as e:
                raise e.__class__("This method requires Astropy to be installed.")

            skycoord = astropy.coordinates.SkyCoord(skycoord).itrs  # Astropy ITRS is GEO
            # Ticktock constructor warns if use_irbem is None; just do right thing
            use_Re = IRBEM_RE if use_irbem else DEFAULTS.values.ellipsoid['A']  # kilometers
            data = (skycoord.cartesian.xyz.to('m').value / (1000 * use_Re)).T  # convert Cartesian to Re units
            ticks = spacepy.time.Ticktock(skycoord.obstime)

            return cls(data, 'GEO', 'car', ticks=ticks, use_irbem=use_irbem)

    def car2sph(car_in):
        """
        Coordinate transformation from Cartesian to spherical

        Parameters
        ----------
            - car_in (list or ndarray) : coordinate points in (n,3) shape with n coordinate points in
                units of [Re, Re, Re] = [x,y,z]

        Returns
        -------
            - results (ndarray) : values after conversion to spherical coordinates in
                radius, latitude, longitude in units of [Re, deg, deg]

        Examples
        --------
        >>> sph2car([1,45,0])
        array([ 0.70710678,  0.        ,  0.70710678])

        See Also
        --------
        sph2car
        """
        if isinstance(car_in[0], numbers.Number):
            car = np.array([car_in])
        else:
            car = np.asanyarray(car_in)

        res = np.zeros(np.shape(car))
        for i in np.arange(len(car)):
            x, y, z = car[i, 0], car[i, 1], car[i, 2]
            r = np.sqrt(x*x+y*y+z*z)
            sq = np.sqrt(x*x+y*y)
            if (x == 0) & (y == 0):  # on the poles
                longi = 0.
                if z < 0:
                    lati = -90.
                else:
                    lati = 90.0
            else:
                longi = np.arctan2(y, x)*180./np.pi
                lati = 90. - np.arctan2(sq, z)*180./np.pi
            res[i, :] = [r, lati, longi]

        if isinstance(car_in[0], numbers.Number):
            return res[0]
        else:
            return res

    def sph2car(sph_in):
        """
        Coordinate transformation from spherical to Cartesian

        Parameters
        ----------
            - sph_in (list or ndarray) : coordinate points in (n,3) shape with n coordinate points in
                units of [Re, deg, deg] = [r, latitude, longitude]

        Returns
        -------
            - results (ndarray) : values after conversion to cartesian coordinates x,y,z

        Examples
        --------
        >>> sph2car([1,45,45])
        array([ 0.5       ,  0.5       ,  0.70710678])

        See Also
        --------
        car2sph
        """
        if isinstance(sph_in[0], numbers.Number):
            sph = np.array([sph_in])
        else:
            sph = np.asanyarray(sph_in)

        res = np.zeros(np.shape(sph))
        for i in np.arange(len(sph)):
            r, lati, longi = sph[i, 0], sph[i, 1], sph[i, 2]
            colat = np.pi/2. - lati*np.pi/180.
            x = r*np.sin(colat)*np.cos(longi*np.pi/180.)
            y = r*np.sin(colat)*np.sin(longi*np.pi/180.)
            z = r*np.cos(colat)
            res[i, :] = [x, y, z]

        if isinstance(sph_in[0], numbers.Number):
            return res[0]
        else:
            return res

    def quaternionNormalize(Qin, scalarPos='last'):
        '''
        Given an input quaternion (or array of quaternions), return the unit quaternion

        Parameters
        ----------
        vec : array_like
            input quaternion to normalize

        Returns
        -------
        out : array_like
            normalized quaternion

        Examples
        --------
        >>> import spacepy.coordinates
        >>> spacepy.coordinates.quaternionNormalize([0.707, 0, 0.707, 0.2])
        array([ 0.69337122,  0.        ,  0.69337122,  0.19614462])
        '''
        w = {'first': 0, 'last': 3}.get(scalarPos.lower())
        if w is None:
            errmsg = 'quaternionNormalize: scalarPos must be set to "First" or "Last"'
            raise NotImplementedError(errmsg)

        Quse = np.asanyarray(Qin).astype(float)
        squeeze = Quse.ndim < 2
        if squeeze:
            Quse = np.expand_dims(Quse, axis=0)
        magn = np.linalg.norm(Quse, axis=-1)
        # Find places where the magnitude is tiny, convert to unit real
        idx = np.where(magn <= 1e-12)
        magn[idx] = 1
        outarr = Quse / np.expand_dims(magn, axis=magn.ndim)
        outarr[idx + (None,)] = 0
        outarr[idx + (w,)] = 1
        if squeeze:
            outarr = outarr[0, ...]
        return outarr

    def quaternionRotateVector(Qin, Vin, scalarPos='last', normalize=True):
        '''
        Given quaternions and vectors, return the vectors rotated by the quaternions

        Parameters
        ----------
        Qin : array_like
            input quaternion to rotate by
        Vin : array-like
            input vector to rotate

        Returns
        -------
        out : array_like
            rotated vector

        Examples
        --------
        >>> import spacepy.coordinates
        >>> import numpy as np
        >>> vec = [1, 0, 0]
        >>> quat_wijk = [np.sin(np.pi/4), 0, np.sin(np.pi/4), 0.0]
        >>> quat_ijkw = [0.0, np.sin(np.pi/4), 0, np.sin(np.pi/4)]
        >>> spacepy.coordinates.quaternionRotateVector(quat_ijkw, vec)
        array([ 0.,  0., -1.])
        >>> spacepy.coordinates.quaternionRotateVector(
        ...     quat_wijk, vec, scalarPos='first')
        array([ 0.,  0., -1.])

        See Also
        --------
        quaternionMultiply
        '''
        if scalarPos.lower() == 'last':
            i, j, k = 0, 1, 2
            w = 3
        elif scalarPos.lower() == 'first':
            i, j, k = 1, 2, 3
            w = 0
        else:
            errmsg = 'quaternionRotateVector: scalarPos must be set to "First" or "Last"'
            raise NotImplementedError(errmsg)

        Quse = np.asanyarray(Qin).astype(float)
        if normalize:
            Quse = quaternionNormalize(Quse, scalarPos=scalarPos)
        Vuse = np.asanyarray(Vin).astype(float)
        try:
            Quse.shape[1]
        except IndexError:
            Quse = np.asanyarray([Quse])
        try:
            Vuse.shape[1]
        except IndexError:
            Vuse = np.asanyarray([Vuse])
        try:
            assert Vuse.shape[0] == Quse.shape[0]
        except AssertionError:
            errmsg = ' '.join(['quaternionRotateVector:',
                               'Input vector array must have same length',
                               'as input quaternion array'])
            raise ValueError(errmsg)

        outarr = np.empty((Quse.shape[0], 3))
        for idx, row in enumerate(Quse):
            Vrow = Vuse[idx]
            ii, jj, kk, ww = row[i]**2, row[j]**2, row[k]**2, row[w]**2
            iw, jw, kw = row[i]*row[w], row[j]*row[w], row[k]*row[w]
            ij, ik, jk = row[i]*row[j], row[i]*row[k], row[j]*row[k]

            outarr[idx, 0] = ww*Vrow[0] + 2*jw*Vrow[2] - 2*kw*Vrow[1] + \
                             ii*Vrow[0] + 2*ij*Vrow[1] + 2*ik*Vrow[2] - \
                             kk*Vrow[0] - jj*Vrow[0]
            outarr[idx, 1] = 2*ij*Vrow[0] + jj*Vrow[1] + 2*jk*Vrow[2] + \
                             2*kw*Vrow[0] - kk*Vrow[1] + ww*Vrow[1] - \
                             2*iw*Vrow[2] - ii*Vrow[1]
            outarr[idx, 2] = 2*ik*Vrow[0] + 2*jk*Vrow[1] + kk*Vrow[2] - \
                             2.0*jw*Vrow[0] - jj*Vrow[2] + 2.0*iw*Vrow[1] - \
                             ii*Vrow[2] + ww*Vrow[2]
        return outarr.squeeze()

    def quaternionMultiply(Qin1, Qin2, scalarPos='last'):
        '''
        Given quaternions, return the product, i.e. Qin1*Qin2

        Parameters
        ----------
        Qin1 : array_like
            input quaternion, first position
        Qin2 : array-like
            input quaternion, second position

        Returns
        -------
        out : array_like
            quaternion product

        Examples
        --------
        >>> import spacepy.coordinates
        >>> import numpy as np
        >>> vecX = [1, 0, 0] #shared X-axis
        >>> vecZ = [0, 0, 1] #unshared, but similar, Z-axis
        >>> quat_eci_to_gsm = [-0.05395384,  0.07589845, -0.15172533,  0.98402634]
        >>> quat_eci_to_gse = [ 0.20016056,  0.03445775, -0.16611386,  0.96496352]
        >>> quat_gsm_to_eci = spacepy.coordinates.quaternionConjugate(
        ...     quat_eci_to_gsm)
        >>> quat_gse_to_gsm = spacepy.coordinates.quaternionMultiply(
        ...     quat_gsm_to_eci, quat_eci_to_gse)
        >>> spacepy.coordinates.quaternionRotateVector(quat_gse_to_gsm, vecX)
        array([  1.00000000e+00,   1.06536725e-09,  -1.16892107e-08])
        >>> spacepy.coordinates.quaternionRotateVector(quat_gse_to_gsm, vecZ)
        array([  1.06802834e-08,  -4.95669027e-01,   8.68511494e-01])
        '''
        if scalarPos.lower() == 'last':
            i, j, k = 0, 1, 2
            w = 3
        elif scalarPos.lower() == 'first':
            i, j, k = 1, 2, 3
            w = 0
        else:
            raise NotImplementedError('quaternionMultiply: scalarPos must be set to "First" or "Last"')

        Quse1 = np.asanyarray(Qin1).astype(float)
        Quse2 = np.asanyarray(Qin2).astype(float)
        try:
            Quse1.shape[1]
        except IndexError:
            Quse1 = np.asanyarray([Quse1])
        try:
            Quse2.shape[1]
        except IndexError:
            Quse2 = np.asanyarray([Quse2])
        try:
            assert Quse2.shape == Quse1.shape
        except AssertionError:
            raise ValueError('quaternionMultiply: Input quaternion arrays must have same length')

        outarr = np.empty_like(Quse1)
        for idx, row in enumerate(Quse1):
            row1 = row
            row2 = Quse2[idx]
            # vector components
            outarr[idx, i] = row1[w]*row2[i] + row1[i]*row2[w] + row1[j]*row2[k] - row1[k]*row2[j]
            outarr[idx, j] = row1[w]*row2[j] - row1[i]*row2[k] + row1[j]*row2[w] + row1[k]*row2[i]
            outarr[idx, k] = row1[w]*row2[k] + row1[i]*row2[j] - row1[j]*row2[i] + row1[k]*row2[w]
            # real part
            outarr[idx, w] = row1[w]*row2[w] - row1[i]*row2[i] - row1[j]*row2[j] - row1[k]*row2[k]
        return outarr.squeeze()

    def quaternionConjugate(Qin, scalarPos='last'):
        '''
        Given an input quaternion (or array of quaternions), return the conjugate

        Parameters
        ----------
        Qin : array_like
            input quaternion to conjugate

        Returns
        -------
        out : array_like
            conjugate quaternion

        Examples
        --------
        >>> import spacepy.coordinates
        >>> spacepy.coordinates.quaternionConjugate(
        ...     [0.707, 0, 0.707, 0.2], scalarPos='last')
        array([-0.707, -0.   , -0.707,  0.2  ])

        See Also
        --------
        quaternionMultiply
        '''
        if scalarPos.lower() == 'last':
            i, j, k = 0, 1, 2
            w = 3
        elif scalarPos.lower() == 'first':
            i, j, k = 1, 2, 3
            w = 0
        else:
            errmsg = 'quaternionConjugate: scalarPos must be set to "First" or "Last"'
            raise NotImplementedError(errmsg)

        Quse = np.asanyarray(Qin).astype(float)
        try:
            Quse.shape[1]
        except IndexError:
            Quse = np.asanyarray([Quse])
        outarr = np.empty_like(Quse)
        for idx, row in enumerate(Quse):
            outarr[idx, i] = -row[i]
            outarr[idx, j] = -row[j]
            outarr[idx, k] = -row[k]
            outarr[idx, w] = row[w]

        return outarr.squeeze()

    def quaternionFromMatrix(matrix, scalarPos='last'):
        '''
        Given an input rotation matrix, return the equivalent quaternion

        The output has one fewer axis than the input (the last axis) and the
        shape is otherwise unchanged, allowing multi-dimensional matrix input.

        Parameters
        ----------
        matrix : array_like
            input rotation matrix or array of matrices

        Returns
        -------
        out : array_like
            Quaternions representing the same rotation as the input rotation
            matrices.

        Other Parameters
        ----------------
        scalarPos : str
            Location of the scalar component of the output quaternion, either
            'last' (default) or 'first'.

        Raises
        ------
        NotImplementedError
            for invalid values of ``scalarPos``

        ValueError
            for inputs which are obviously not valid 3D rotation matrices or
            arrays thereof: if the size doesn't end in (3, 3), if the matrix is
            not orthogonal, or not a proper rotation.

        See Also
        --------
        quaternionToMatrix

        Notes
        -----
        .. versionadded:: 0.2.2

        No attempt is made to resolve the sign ambiguity; in particular,
        conversions of very similar matrices may result in equivalent quaternions
        with the opposite sign. This may have implications for interpolating a
        sequence of quaternions.

        The conversion of a rotation matrix to a quaternion suffers from some
        of the same disadvantages inherent to rotation matrices, such as potential
        numerical instabilities. Working in quaternion space as much as possible
        is recommended.

        There are several algorithms; the most well-known algorithm for this
        conversion is Shepperd's [#Shepperd]_, although the many "rediscoveries"
        indicate
        it is not sufficiently well-known. This function uses the method of
        Bar-Itzhack [#BarItzhack]_ (version 3), which should be resistant to small
        errors in the rotation matrix. As a result, the input checking is quite
        coarse and will likely accept many matrices that do not represent valid
        rotations.

        Also potentially of interest, although not implemented here, is Sarabandi
        and Thomas [#Sarabandi]_.

        References
        ----------
        .. [#Shepperd] S.W. Shepperd, "Quaternion from rotation matrix," Journal of
                Guidance and Control, Vol. 1, No. 3, pp. 223-224, 1978,
                `doi:10.2514/3.55767b <https://doi.org/10.2514/3.55767b>`_

        .. [#BarItzhack] I. Y. Bar-Itzhack, "New method for extracting the
                quaternion from a rotation matrix", AIAA Journal of Guidance,
                Control and Dynamics, 23 (6): 1085–1087,
                `doi:10.2514/2.4654 <https://doi.org/10.2514/2.4654>`_

        .. [#Sarabandi] S. Sarabandi and F. Thomas, "Accurate Computation of
                Quaternions from Rotation Matrices", In: Lenarcic J.,
                Parenti-Castelli V. (eds) Advances in Robot Kinematics 2018,
                Springer.
                `doi:10.1007/978-3-319-93188-3_5
                <https://doi.org/10.1007/978-3-319-93188-3_5>`_

        Examples
        --------
        >>> import spacepy.coordinates
        >>> spacepy.coordinates.quaternionFromMatrix(
        ...     [[ 0.,  0.,  1.],
        ...      [ 1.,  0.,  0.],
        ...      [ 0.,  1.,  0.]])
        array([0.5, 0.5, 0.5, 0.5])
        '''
        if scalarPos.lower() not in ('last', 'first'):
            raise NotImplementedError(
                'quaternionFromMatrix: scalarPos must be set to "First" or "Last"')
        matrix = np.asanyarray(matrix)
        if len(matrix.shape) < 2 or matrix.shape[-2:] != (3, 3):
            raise ValueError(
                'Input does not appear to be 3D rotation matrix, wrong size.')
        for i in np.ndindex(matrix.shape[:-2]):
            m = matrix[i + (Ellipsis,)]
            if not np.allclose(np.dot(m, m.transpose()),
                               np.identity(3), atol=0.2):
                raise ValueError('Input rotation matrix{} not orthogonal.'.format(
                    '' if len(matrix.shape) == 2 else ' at ' + str(i)))
            det = np.linalg.det(m)
            if det < 0:
                raise ValueError('Input rotation matrix at {} not proper.'
                                 .format(str(i)))
        inshape = matrix.shape
        # Flatten out most dimensions, easier to index
        matrix = np.reshape(matrix, (-1, 3, 3))
        # Indexing in Bar-Itzhack is reversed relative to numpy
        k = (matrix[..., 0, 0] - matrix[..., 1, 1] - matrix[..., 2, 2],
             matrix[..., 0, 1] + matrix[..., 1, 0],
             matrix[..., 0, 2] + matrix[..., 2, 0],
             matrix[..., 2, 1] - matrix[..., 1, 2],  # row 0
             matrix[..., 0, 1] + matrix[..., 1, 0],
             matrix[..., 1, 1] - matrix[..., 0, 0] - matrix[..., 2, 2],
             matrix[..., 1, 2] + matrix[..., 2, 1],
             matrix[..., 0, 2] - matrix[..., 2, 0],  # row 1
             matrix[..., 0, 2] + matrix[..., 2, 0],
             matrix[..., 1, 2] + matrix[..., 2, 1],
             matrix[..., 2, 2] - matrix[..., 0, 0] - matrix[..., 1, 1],
             matrix[..., 1, 0] - matrix[..., 0, 1],  # row 2
             matrix[..., 2, 1] - matrix[..., 1, 2],
             matrix[..., 0, 2] - matrix[..., 2, 0],
             matrix[..., 1, 0] - matrix[..., 0, 1],
             matrix[..., 0, 0] + matrix[..., 1, 1] + matrix[..., 2, 2]  # row 3
             )
        # Stack together on a new axis after the input sample, then
        # split that axis into the 4x4 k arrays
        k = np.reshape(np.stack(k, axis=1), (-1, 4, 4))
        k = k / 3.
        evals, evects = np.linalg.eig(k)
        evects = np.real(evects)
        idx = np.argmax(evals, axis=-1)
        Qout = evects[np.mgrid[0:len(idx)], :, idx]
        if scalarPos.lower() == 'first':
            Qout = np.roll(Qout, 1, axis=-1)
        # Recover original shape
        Qout = np.reshape(Qout, inshape[:-2] + (4,))
        return Qout

    def quaternionToMatrix(Qin, scalarPos='last', normalize=True):
        '''
        Given an input quaternion, return the equivalent rotation matrix.

        The output has one more axis than the input (the last axis) and the
        shape is otherwise unchanged, allowing multi-dimensional quaternion input.

        Parameters
        ----------
        Qin : array_like
            input quaternion or array of quaternions, must be normalized.

        Returns
        -------
        out : array_like
            Rotation matrix

        Other Parameters
        ----------------
        scalarPos : str
            Location of the scalar component of the input quaternion, either
            'last' (default) or 'first'.

        normalize : True
            Normalize input quaternions before conversion (default). If False,
            raises error for non-normalized.

        Raises
        ------
        NotImplementedError
            for invalid values of ``scalarPos``.

        ValueError
            for inputs which are not valid normalized quaternions or arrays
            thereof: if the size doesn't end in (4), if the quaternion is not
            normalized and ``normalize`` is False.

        See Also
        --------
        quaternionFromMatrix

        Notes
        -----
        .. versionadded:: 0.2.2

        Implementation of the Euler–Rodrigues formula.

        Examples
        --------
        >>> import spacepy.coordinates
        >>> spacepy.coordinates.quaternionToMatrix([0.5, 0.5, 0.5, 0.5])
        array([[ 0.,  0.,  1.],
               [ 1.,  0.,  0.],
               [ 0.,  1.,  0.]])
        '''
        if scalarPos.lower() not in ('last', 'first'):
            raise NotImplementedError(
                'quaternionToMatrix: scalarPos must be set to "First" or "Last"')
        Qin = np.asanyarray(Qin)
        if Qin.shape[-1] != 4:
            raise ValueError('Input does not appear to be quaternion, wrong size.')
        if normalize:
            Qin = quaternionNormalize(Qin, scalarPos=scalarPos)
        if scalarPos.lower() == 'first':
            Qin = np.roll(Qin, -1, axis=-1)
        if not np.allclose(np.sum(Qin ** 2, axis=-1), 1):
            raise ValueError('Input quaternion not normalized.')
        # Maintain dimensions at end for stacking
        a, b, c, d = Qin[..., -1:], Qin[..., 0:1], Qin[..., 1:2], Qin[..., 2:3]
        # Output array, "flattened"
        # This can probably be written with a clever vector notation...
        out = np.concatenate((a ** 2 + b ** 2 - c ** 2 - d ** 2,
                              2 * (b * c - a * d),
                              2 * (b * d + a * c),
                              2 * (b * c + a * d),
                              a ** 2 + c ** 2 - b ** 2 - d ** 2,
                              2 * (c * d - a * b),
                              2 * (b * d - a * c),
                              2 * (c * d + a * b),
                              a ** 2 + d ** 2 - b ** 2 - c ** 2,
                              ), axis=-1)
        # And last two dims are 3x3 array for matrix
        out = np.reshape(out, out.shape[:-1] + (3, 3))
        return out