# Source code for spacepy.ctrans

    """CTrans: Module for backend coordinate transformations in SpacePy

    This module is primarily intended to provide a backend for the standard
    :class:`~spacepy.coordinates.Coords` class rather than direct use, and the
    interface is subject to change.

    The :class:`CTrans` class calculates all of the necessary information
    to convert between different coordinate systems *at a single time*. By
    using :class:`~spacepy.coordinates.Coords` the handling of multiple
    times is built in, and the calling syntax is backwards compatible with
    the legacy IRBEM-backed coordinate transforms.

    Coordinate systems supported by this module can broadly be described
    by two categories. The first category is a broad set of Earth-centered
    coordinate systems that are specified by astronomical parameters.
    If we consider the International Celestial Reference Frame to be our
    starting point, then taking the origin as the center of the Earth
    instead of the solar barycenter gives us the Geocentric Celestial
    Reference Frame (GCRF). All coordinate systems described here are
    right-handed Cartesian systems, except geodetic.

    Systems and their relationships:

    - ECI2000: Earth-Centered Inertial, J2000 epoch
        This system can be considered equivalent to the GCRF, to within 10s
        of milliarcseconds. The z-axis is aligned with the mean celestial pole
        at the J2000 epoch. The x-axis is aligned with the mean equinox at the
        J2000 epoch. The y-axis completes and lies in the plane of the
        celestial equator.
    - ECIMOD: Earth-Centered Inertial, Mean-of-Date
        This system accounts for precession between the J2000 epoch and the
        date of interest: The coordinate system is time-dependent. The system
        is defined similarly to ECI2000, but uses the mean equinox and mean
        equator of the date of interest.
    - ECITOD: Earth-Centered Inertial, True-of-Date
        This system builds on ECIMOD and accounts for the nutation (the short-
        period perturbations on the precession). This system is therefore
        considered to use the true equator and true equinox of date.
    - TEME: Earth-Centered Inertial, True Equator Mean Equinox
        This system is poorly defined in the literature, despite being used in
        the SGP4 orbital propagator (note that multiple versions of SGP4 exist,
        see e.g. Vallado et al. 2006; AIAA 2006-6753-Rev2). The mean equinox here
        is not the same as the mean equinox used in, e.g., ECIMOD, but lies along
        the true equator between the origin of the Pseudo Earth Fixed and ECITOD
        frames. It is highly recommended that TEME coordinates are converted to a
        standard system (e.g., ECI2000) before passing to other users or to
        different software.
    - GSE: Geocentric Solar Ecliptic
        This system is not inertial. It is Earth-centered, with the x-axis
        pointing towards the Sun. The y-axis lies in the mean ecliptic plane
        of date, pointing in the anti-orbit direction. The z-axis is parallel
        to the mean ecliptic pole.
    - GEO: Geocentric Geographic
        This system is not inertial. It is Earth-Centered and Earth-Fixed (also
        called ECEF), so that the coordinates of a point fixed on (or relative
        to) the surface of the Earth do not change. The x-axis lies in the
        Earth's equatorial plane (zero latitude) and intersects the Prime
        Meridian (zero longitude; Greenwich, UK). The z-axis points to True
        North (which is roughly aligned with the instantaneous rotation axis).
    - GDZ: Geodetic
        This system is not inertial and is defined in terms of altitude above
        a reference ellipsoid, the geodetic latitude, and geodetic longitude.
        Geodetic longitude is identical to GEO longitude. Both the altitude
        and latitude depend on the ellipsoid used. While geodetic latitude is
        close to geographic latitude, they are not the same. The default here is
        to use the WGS84 reference ellipsoid.

    The remaining coordinate systems are also reference to Earth's magnetic field.
    Different versions of these systems exist, but the most common (and those given
    here) use a centered dipole axis.

    - GSM: Geocentric Solar Magnetospheric
        This system is similar to GSE, but is defined such that the centered dipole
        lies in the x-z plane. As in all of these systems, z is positive northward.
        The y-axis is thus perpendicular to both the Sun-Earth line and the
        centered dipole axis (of date, defined using the first 3 coefficients of the
        IGRF/DGRF). GSM is therefore a rotation about the x-axis from the GSE system.
    - SM: Solar Magnetic
        The z-axis is aligned with the centered dipole axis of date (positive
        northward), and the y-axis is perpendicular to both the Sun-Earth line and
        the dipole axis. As with GSE and GSM, y is positive in the anti-orbit
        direction. The x-axis therefore is not aligned with the Sun-Earth line and
        SM is a rotation about the y-axis from the GSM system.
    - CDMAG: Geomagnetic
        This is a geomagnetic analog of the GEO system. The z-axis is aligned with
        the centered dipole axis of date. The y-axis is perpendicular to
        to both the dipole axis and True North, i.e., y is the cross product of
        the z-axis of the GEO system with the dipole axis. The x-axis completes.

    Classes
    -------
    .. autosummary::
        :template: clean_class.rst
        :toctree: autosummary

        CTrans
        Ellipsoid

    Functions
    ---------
    .. autosummary::
        :template: clean_function.rst
        :toctree: autosummary

        convert_multitime
        gdz_to_geo
        geo_to_gdz
        geo_to_rll
        rll_to_geo

    Submodules
    ----------
    .. autosummary::
        :template: clean_module.rst
        :toctree: autosummary

        iau80n
    """

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

    import datetime as dt
    import collections
    from math import fmod

    import numpy as np
    import scipy.constants

    from spacepy import datamodel as dm
    from spacepy import time as spt
    from spacepy import igrf

    class Ellipsoid(dm.SpaceData):
        """Ellipsoid definition class for geodetic coordinates

        Other Parameters
        ----------------
        name : str
            Name for ellipsoid, stored in attrs of returned Ellipsoid instance.
            Default is 'WGS84'
        A : float
            Semi-major axis (equatorial radius) of ellipsoid in km.
            Default is 6378.137km (WGS84_A)
        iFlat : float
            Inverse flattening of ellipsoid. Default is WGS84 value
            of 298.257223563.

        Returns
        -------
        out : Ellipsoid
            Ellipsoid instance storing all relevant paramters for geodetic conversion

        Notes
        -----

        .. versionadded:: 0.3.0
        """
        def __init__(self, name='WGS84', A=6378.137, iFlat=298.257223563):
            super(Ellipsoid, self).__init__(A=A, iFlat=iFlat, attrs={'name': name})
            self['A2'] = self['A']**2  # [km^2]
            self['Flat'] = 1/self['iFlat']
            self['E2'] = 2*self['Flat'] - self['Flat']**2
            self['E4'] = self['E2']*self['E2']
            self['1mE2'] = 1 - self['E2']
            self['B'] = self['A']*np.sqrt(self['1mE2'])
            self['B2'] = self['B']**2  # [km^2]
            self['A2mB2'] = self['A2'] - self['B2']  # [km^2]
            self['EP2'] = self['A2mB2']/self['B2']  # 2nd eccentricity squared

    # World Geodetic System 1984 (WGS84) parameters
    WGS84 = Ellipsoid()

    magsys = ['GSM', 'SM', 'CDMAG']

    class CTrans(dm.SpaceData):
        """Coordinate transformation class for a single instance in time

        A general coordinate conversion routine, which takes a numpy array (Nx3)
        of Cartesian vectors along with the names of the input and output
        coordinate systems and returns an array of the converted coordinates.

        Parameters
        ----------
        ctime : (spacepy.time.Ticktock, datetime, float, string)
            Input time stamp. Must have one time only. Accepted input formats

        Returns
        -------
        out : CTrans
            instance with self.convert, etc.

        Other Parameters
        ----------------
        ephmodel : str, optional
            Select ephemerides model (e.g., for determining Sun direction).
            Currently only 'LGMDEFAULT' is supported, for consistency with
            LANLGeoMag implementation.
        pnmodel : str, optional
            Select precession/nutation model set. Options are: 'IAU82' (default),
            and 'IAU00'.
        eop : bool, optional
            Use Earth Orientation Parameters

        See Also
        --------
        spacepy.coordinates.Coords

        Notes
        -----

        .. versionadded:: 0.3.0

        .. rubric:: Methods

        .. autosummary::

            ~CTrans.calcTimes
            ~CTrans.calcOrbitParams
            ~CTrans.calcCoreTransforms
            ~CTrans.calcMagTransforms
            ~CTrans.convert
            ~CTrans.getEOP
            ~CTrans.gmst

        .. automethod:: calcTimes
        .. automethod:: calcOrbitParams
        .. automethod:: calcCoreTransforms
        .. automethod:: calcMagTransforms
        .. automethod:: convert
        .. automethod:: getEOP
        .. automethod:: gmst
        """
        def __init__(self, ctime, ephmodel=None, pnmodel=None, eop=False):
            super(CTrans, self).__init__()
            if ephmodel is not None:
                self._raiseErr(NotImplementedError, 'ephmodel')
            else:
                self.attrs['ephmodel'] = 'LGMDEFAULT'
            if pnmodel is not None:
                if pnmodel not in ['IAU82', 'IAU00']:
                    self._raiseErr(NotImplementedError, 'pnmodel')
                self.attrs['pnmodel'] = pnmodel
            else:
                self.attrs['pnmodel'] = 'IAU82'

            if isinstance(ctime, spt.Ticktock):
                # input time is ticktock
                if len(ctime.data) != 1:
                    # Input time is Ticktock, but has a length > 1
                    self._raiseErr(ValueError, 'time_in')
            elif isinstance(ctime, dt.datetime):
                # Input time is datetime
                ctime = spt.Ticktock(ctime, dtype='UTC')
            else:
                try:
                    if len(ctime) == 1:
                        ctime = ctime[0]
                        ctime = spt.Ticktock(ctime)  # Guess dtype
                    else:
                        self._raiseErr(TypeError, 'time_in')
                except TypeError:
                    self._raiseErr(TypeError, 'time_in')
            self.attrs['time'] = ctime

            # Make key information immutable, but referenced by name
            self._factory = dict()
            self._factory['constants'] = collections.namedtuple('Constants',
                                                                'twopi, j2000_jd, j1900_jd, ' +
                                                                'j1990_jd, daysec, daycentury, ' +
                                                                'arcsec, AU, Re',
                                                                )
            self._factory['eop'] = collections.namedtuple('EarthOrientationParameters',
                                                          'DUT1, xp, yp, ddPsi, ddEps',
                                                          )
            self._setconstants()
            self.getEOP(useEOP=eop)
            self.__status = {'time': False,
                             'orbital': False,
                             'transformCore': False,
                             'transformMag': False,
                             'timeInProgress': False,
                             'coreInProgress': False,
                             'useEOP': eop
                             }

        def _setconstants(self):
            """Set constants to be used in calculations

            Constants set through this method are immutable to ensure
            consistency in calculations.
            """
            self['constants'] = self._factory['constants'](twopi=scipy.constants.pi*2,
                                                           j2000_jd=2451545.0,
                                                           j1990_jd=2447891.5,
                                                           j1900_jd=2415020.0,
                                                           daysec=86400,
                                                           daycentury=36525.0,
                                                           arcsec=scipy.constants.arcsec,
                                                           AU=scipy.constants.au/1e3,  # 1AU in km
                                                           Re=WGS84['A'],  # WGS84_A radius in km
                                                           )

        def getEOP(self, useEOP=False):
            """Get/set Earth Orientation Parameters

            Parameters
            ----------
            useEOP : bool
                If True, use Earth Orientation Parameters. Default False.

            Notes
            -----
            Currently Earth Orientation Parameters are all set to zero.
            Use is not yet supported.
            """
            if not useEOP:
                # DUT1 in seconds
                # xp, yp in arcseconds
                # ddPsi and ddEps in arcseconds
                self['EarthOrientationParameters'] = self._factory['eop'](DUT1=0,
                                                                          xp=0,
                                                                          yp=0,
                                                                          ddPsi=0,
                                                                          ddEps=0)
            else:
                # Note: These case still be set manually by directly calling the
                # factory method.
                self._raiseErr(NotImplementedError, 'eop')

        def calcTimes(self, recalc=False, **kwargs):
            """Calculate time in systems required to set up coordinate transforms

            Sets Julian Date and Julian centuries in UTC, TAI, UT1, and TT systems.

            Parameters
            ----------
            recalc : bool, optional
                If True, recalculate the times for coordinate transformation. Default
                is False.
            """
            if self.__status['time'] and not recalc:
                return
            # Set up various time systems required and variable shortcuts
            eop = self['EarthOrientationParameters']
            const = self['constants']
            ctime = self.attrs['time']

            # UT1
            self['UTC'] = ctime
            self['UTC_JD'] = ctime.JD[0]
            self['UTC_JC'] = (self['UTC_JD'] - const.j2000_jd)/const.daycentury
            self['TAI_JD'] = spt._days1958(self['UTC'].TAI[0], leaps='continuous') + 2436205.0
            if self.__status['useEOP']:
                self['UT1'] = spt.Ticktock(ctime.UTC[0] + dt.timedelta(seconds=eop.DUT1), dtype='UTC')
                self['UT1_JD'] = self['UT1'].JD[0]
            else:
                # no EOP, so UT1 = UTC and we can save time recalculating times
                self['UT1'] = self['UTC']
                self['UT1_JD'] = self['UTC_JD']
            self['UT1_JC'] = (self['UT1_JD'] - const.j2000_jd)/const.daycentury

            # Terrestrial Time in Julian centuries since J2000
            self['TT_JD'] = self['TAI_JD'] + 32.184/const.daysec
            self['TT_JC'] = (self['TT_JD'] - const.j2000_jd)/const.daycentury

            # Greenwich Mean Sidereal Time
            if 'from_gmst' not in kwargs:
                self.__status['timeInProgress'] = True
                self.gmst()
                self.__status['timeInProgress'] = False
            self.__status['time'] = True

        def calcOrbitParams(self, recalc=False):
            """Calculate Earth orbit parameters needed for coordinate transforms

            Calculates  Earth's orbital parameters required for defining coordinate
            system transformations, such as orbital eccentricity, the obliquity of
            the ecliptic, anomalies, and precession angles.

            Parameters
            ----------
            recalc : bool, optional
                If True, recalculate the orbital parameters for coordinate transformation.
                Default is False.
            """
            if not (self.__status['time'] or recalc):
                self.calcTimes(recalc=recalc)
            if self.__status['orbital'] and not recalc:
                # Don't recalculate unless it's asked for
                return
            const = self['constants']

            # Julian centuries with 1900 reference epoch
            self['JulianCenturies'] = (self['TT_JD'] - const.j1900_jd)/const.daycentury
            TU = self['JulianCenturies']  # since 1900
            TU2 = TU*TU

            # Orbital parameters required for later use
            # self['MeanEclipticLongitude'] = np.deg2rad(279.6966778 + 36000.76892*TU
            #                                            + 0.0003025*TU2)
            # self['EclipticLongitudePerigee'] = np.deg2rad(281.2208444 + 1.719175*TU
            #                                               + 0.000452778*TU2)
            self['Eccentricity'] = 0.01675104 - 0.0000418*TU - 0.000000126*TU2
            TT = self['TT_JC']  # in Julian century (J2000) format
            TT2 = TT*TT
            self['ObliquityEcliptic'] = 84381.448 - 46.8150*TT\
                - 0.00059*TT2 + 0.001813*TT2*TT
            self['ObliquityEcliptic_rad'] = self['ObliquityEcliptic']*const.arcsec
            self['ObliquityEcliptic'] = self['ObliquityEcliptic']

            # Anomalies
            days_since_1990 = self['TT_JD'] - const.j1990_jd
            varep90 = np.deg2rad(279.403303)
            varpi90 = np.deg2rad(282.768422)
            M = fmod((const.twopi/365.242191*days_since_1990), const.twopi)
            self['MeanAnomaly'] = fmod((M + varep90 - varpi90), const.twopi)
            ecc_anom = self._kepler(self['MeanAnomaly'], self['Eccentricity'])
            self['TrueAnomaly'] = 2.0*np.arctan(np.sqrt((1.0+self['Eccentricity'])
                                                / (1.0-self['Eccentricity']))*np.tan(ecc_anom/2.0))
            self['LambdaSun'] = fmod((self['TrueAnomaly'] + varpi90), const.twopi)
            self['OmegaMoon'] = fmod((125.04455501 - (5*360 + 134.1361851)*TT
                                     + 0.0020756*TT2 + 2.139e-6*TT2*TT), 360.0)  # degrees

            # Precession angles in degrees
            self['PrecessionZeta'] = 0.6406161*TT + 0.0000839*TT2 + 5.0e-6*TT2*TT
            self['PrecessionZee'] = 0.6406161*TT + 0.0003041*TT2 + 5.1e-6*TT2*TT
            self['PrecessionTheta'] = 0.5567530*TT - 0.0001185*TT2 - 1.16e-5*TT2*TT

            # Nutation terms
            self._calcNutation()
            self.__status['orbital'] = True

        def _calcNutation(self, nTerms=106):
            """Calculate nutation-related quantities

            Calculates nutation parameters, equation of equinoxes,
            true obliquity of the ecliptic, and updates object values
            """
            # Get dPSi and dEps in arcsec
            dPsi, dEps = self._nutation(nTerms=nTerms)  # default 106-term nutation series
            # Now apply EOP corrections
            dPsi += self['EarthOrientationParameters'].ddPsi
            dEps += self['EarthOrientationParameters'].ddEps
            # Equation of the equinoxes
            omegamoon_rad = np.deg2rad(self['OmegaMoon'])
            EquatEquin = dPsi*np.cos(self['ObliquityEcliptic_rad']) \
                + 0.00264*np.sin(omegamoon_rad) \
                + 0.000063*np.sin(2.0*omegamoon_rad)  # arcsec
            self['EquatEquin'] = EquatEquin
            # true obliquity of ecliptic
            self['ObliquityTrue'] = dEps + self['ObliquityEcliptic']  # arcsec
            self['ObliquityTrue_rad'] = self['ObliquityTrue']*scipy.constants.arcsec
            self['dPsi'] = dPsi
            self['dEps'] = dEps

        @staticmethod
        def _normVec(vec):
            """Normalize input vector"""
            tmpmag = np.linalg.norm(vec)
            vec /= tmpmag

        def _nutation(self, nTerms=106):
            """Calculate nutation terms dPsi and dEps"""
            pnmodel = self.attrs['pnmodel'].upper()
            if not pnmodel == 'IAU82':
                import warnings
                warnings.warn('Only the IAU1980 nutation model is currently implemented. '
                              + 'This will be used with the selected precession model.')
            else:
                from . import iau80n
                dPsi, dEps = iau80n.nutation(self['TT_JC'], self['constants'], nTerms)
            # TODO: look at implementing IAU2000 as well
            #       See p88 onwards of USNO circular 179, for the coefficients.
            #       That has 1365 terms, and we'll likely want to have the default number evaluated
            #       set much lower.
            #       The equations are p45-48 of the same reference
            return dPsi, dEps

        def calcCoreTransforms(self, recalc=False):
            """
            Calculate core coordinate transform matrices

            These coordinate systems do not require information about
            Earth's magnetic field.
            The systems are:
            Earth-Centered Inertial, J2000 (ECI2000)
            Earth-Centered Inertial, Mean-of-date (ECIMOD)
            Earth-Centered Inertial, True-of-date (ECITOD)
            Geocentric Solar Ecliptic (GSE)
            Geocentric Geographic (GEO)

            Parameters
            ----------
            recalc : bool, optional
                If True, recalculate the core (non-magnetic) coordinate transformations.
                Default is False.
            """
            if not self.__status['orbital'] or recalc:
                self.calcOrbitParams(recalc=recalc)
            if not recalc and self.__status['transformCore']:
                return
            self['Transform'] = dm.SpaceData()
            # ECI J2000 to ECI Mean of Date, and inverse
            zeta = np.deg2rad(self['PrecessionZeta'])
            cosZeta, sinZeta = np.cos(zeta), np.sin(zeta)
            zee = np.deg2rad(self['PrecessionZee'])
            cosZee, sinZee = np.cos(zee), np.sin(zee)
            theta = np.deg2rad(self['PrecessionTheta'])
            cosTheta, sinTheta = np.cos(theta), np.sin(theta)
            eci2000_ecimod = np.empty((3, 3))
            eci2000_ecimod[0, 0] = cosZeta*cosTheta*cosZee-sinZeta*sinZee
            eci2000_ecimod[0, 1] = -sinZeta*cosTheta*cosZee-cosZeta*sinZee
            eci2000_ecimod[0, 2] = -sinTheta*cosZee
            eci2000_ecimod[1, 0] = cosZeta*cosTheta*sinZee+sinZeta*cosZee
            eci2000_ecimod[1, 1] = -sinZeta*cosTheta*sinZee+cosZeta*cosZee
            eci2000_ecimod[1, 2] = -sinTheta*sinZee
            eci2000_ecimod[2, 0] = cosZeta*sinTheta
            eci2000_ecimod[2, 1] = -sinZeta*sinTheta
            eci2000_ecimod[2, 2] = cosTheta
            self['Transform']['ECI2000_ECIMOD'] = eci2000_ecimod
            self['Transform']['ECIMOD_ECI2000'] = eci2000_ecimod.T

            # ECI Mean of Date to ECI True of Date, and inverse
            ecitod_ecimod = np.empty((3, 3))
            dpsi_rad = self['dPsi']*self['constants'].arcsec
            sdps, cdps = np.sin(dpsi_rad), np.cos(dpsi_rad)
            sobl, cobl = np.sin(self['ObliquityEcliptic_rad']), np.cos(self['ObliquityEcliptic_rad'])
            sobt, cobt = np.sin(self['ObliquityTrue_rad']), np.cos(self['ObliquityTrue_rad'])
            ecitod_ecimod[0, 0] = cdps
            ecitod_ecimod[0, 1] = sdps*cobt
            ecitod_ecimod[0, 2] = sobt*sdps
            ecitod_ecimod[1, 0] = -sdps*cobl
            ecitod_ecimod[1, 1] = cobt*cdps*cobl + sobt*sobl
            ecitod_ecimod[1, 2] = sobt*cdps*cobl - sobl*cobt
            ecitod_ecimod[2, 0] = -sdps*sobl
            ecitod_ecimod[2, 1] = cobt*cdps*sobl - sobt*cobl
            ecitod_ecimod[2, 2] = sobt*sobl*cdps + cobt*cobl
            # TODO: if scipy version constraints allow, consider scipy.spatial.transform rotations
            self['Transform']['ECITOD_ECIMOD'] = ecitod_ecimod
            self['Transform']['ECIMOD_ECITOD'] = ecitod_ecimod.T

            # Need to get Sun vector for subsequent transforms, but Sun vector requires
            # J2000 <-> MOD transforms to be set up already
            self.__status['coreInProgress'] = True
            self._calcSun()
            self.__status['coreInProgress'] = False

            # ECI Mean of Date to Geocentric Solar Ecliptic, and reverse
            # First define axes for GSE coordinate system
            self['EclipticPole'] = np.array([0,  # x-hat
                                             -np.sin(self['ObliquityEcliptic_rad']),  # y-hat
                                             np.cos(self['ObliquityEcliptic_rad'])])
            gse_y_vec = np.cross(self['EclipticPole'], self['SunVector_MOD'])
            self._normVec(gse_y_vec)
            gse_z_vec = np.cross(self['SunVector_MOD'], gse_y_vec)
            # Set up transformation matrices for MOD <-> GSE
            ecimod_gse = np.empty((3, 3))
            ecimod_gse[0, :] = self['SunVector_MOD'][...]
            ecimod_gse[1, :] = gse_y_vec[...]
            ecimod_gse[2, :] = gse_z_vec[...]

            self['Transform']['ECIMOD_GSE'] = ecimod_gse
            self['Transform']['GSE_ECIMOD'] = ecimod_gse.T

            # Pseudo Earth Fixed (PEF) to/from ECI TOD
            # Use PEF as a pass-through to get to GEO
            # See "Revisiting Spacetrack Report #3" [Vallado]

            # First, get apparent sidereal time
            # Uses GMST in degrees and Equation of Equinoxes in degrees (IAU82? Or general?)
            gast_deg = self['GMST']*15.0 + self['EquatEquin']/3600.0
            gast_rad = np.deg2rad(gast_deg)
            csga = np.cos(gast_rad)
            snga = np.sin(gast_rad)
            ecitod_pef = np.zeros((3, 3))
            ecitod_pef[0, 0] = csga
            ecitod_pef[0, 1] = snga
            ecitod_pef[1, 0] = -snga
            ecitod_pef[1, 1] = csga
            ecitod_pef[2, 2] = 1.0

            # TEME (SGP4 coordinate system)
            sn = np.sin(self['GMST_rad'])
            cs = np.cos(self['GMST_rad'])
            teme_pef = np.zeros((3, 3))
            teme_pef[0, 0] = cs
            teme_pef[0, 1] = sn
            teme_pef[1, 0] = -sn
            teme_pef[1, 1] = cs
            teme_pef[2, 2] = 1.0

            # add TEME to transforms
            pef_teme = teme_pef.T
            ecitod_teme = pef_teme.dot(ecitod_pef)
            ecimod_teme = ecitod_teme.dot(self['Transform']['ECIMOD_ECITOD'])
            self['Transform']['ECIMOD_TEME'] = ecimod_teme
            self['Transform']['TEME_ECIMOD'] = ecimod_teme.T

            # GEO (Geocentric geographic) to/from PEF (Psuedo Earth Fixed)
            xprad = self['EarthOrientationParameters'].xp*self['constants'].arcsec
            yprad = self['EarthOrientationParameters'].yp*self['constants'].arcsec
            sxp = np.sin(xprad)
            cxp = np.cos(xprad)
            syp = np.sin(yprad)
            cyp = np.cos(yprad)
            pef_geo = np.zeros((3, 3))
            pef_geo[0, 0] = cxp
            pef_geo[0, 1] = sxp*syp
            pef_geo[0, 2] = sxp*cyp
            pef_geo[1, 1] = cyp
            pef_geo[1, 2] = -syp
            pef_geo[2, 0] = -sxp
            pef_geo[2, 1] = cxp*syp
            pef_geo[2, 2] = cxp*cyp

            # make/store ECIMOD<->GEO transforms
            ecitod_geo = pef_geo.dot(ecitod_pef)
            ecimod_geo = ecitod_geo.dot(self['Transform']['ECIMOD_ECITOD'])
            self['Transform']['ECIMOD_GEO'] = ecimod_geo
            self['Transform']['GEO_ECIMOD'] = ecimod_geo.T

            self.__status['transformCore'] = True

        def calcMagTransforms(self, recalc=False):
            """Calculate geophysical coordinate systems

            Calculate transforms for coordinate systems requiring magnetic field
            information.

            These are:
            Solar Magnetic (SM)
            Geocentric Solar Magnetospheric (GSM)
            Geomagnetic, centered dipole (CDMAG)

            Parameters
            ----------
            recalc : bool, optional
                If True, recalculate the core (non-magnetic) coordinate transformations.
                Default is False.
            """
            if not self.__status['transformCore'] or recalc:
                self.calcCoreTransforms()

            # Call IGRF to calculate centered dipole
            self['IGRF'] = igrf.IGRF()
            self['IGRF'].initialize(self['UTC'].UTC[0])

            # Get dipole axis unit vector
            glon = self['IGRF'].dipole['cd_glon_rad']
            gcolat = self['IGRF'].dipole['cd_gcolat_rad']
            dip = np.empty(3)
            dip[0] = np.cos(glon)*np.sin(gcolat)
            dip[1] = np.sin(glon)*np.sin(gcolat)
            dip[2] = np.cos(gcolat)

            # Convert dipole axis to MOD
            dipmod = np.empty(3)
            dipmod[0] = np.cos(self['GMST_rad'])*dip[0] - np.sin(self['GMST_rad'])*dip[1]
            dipmod[1] = np.sin(self['GMST_rad'])*dip[0] + np.cos(self['GMST_rad'])*dip[1]
            dipmod[2] = dip[2]

            # Compute Ygsm axis in MOD
            gsm_y_vec = np.cross(dipmod, self['SunVector_MOD'])
            self._normVec(gsm_y_vec)

            # Compute Zgsm axis in MOD system
            gsm_z_vec = np.cross(self['SunVector_MOD'], gsm_y_vec)

            # Set up transformation matrices for MOD <-> GSM
            ecimod_gsm = np.empty((3, 3))
            ecimod_gsm[0, :] = self['SunVector_MOD'][...]
            ecimod_gsm[1, :] = gsm_y_vec[...]
            ecimod_gsm[2, :] = gsm_z_vec[...]
            self['Transform']['ECIMOD_GSM'] = ecimod_gsm
            self['Transform']['GSM_ECIMOD'] = ecimod_gsm.T

            # Compute the Dipole Tilt angle (psi). First convert dipmod into GSM.
            # Then psi is just the angle between dipgsm and Zgsm
            dipgsm = self.convert(dipmod, 'ECIMOD', 'GSM')
            psidot = np.array([0, 0, 1]).dot(dipgsm)
            self._normVec(psidot)
            psi = np.arccos(psidot)
            psi *= -1 if (dipgsm[0] < 0) else 1
            self['DipoleTilt_rad'] = psi
            self['DipoleTilt'] = np.rad2deg(psi)
            cos_psi = np.cos(psi)
            sin_psi = np.sin(psi)

            # Set up transformation matrices for GSM <-> SM
            gsm_sm = np.empty((3, 3))
            gsm_sm[:, 0] = np.array([cos_psi, 0, sin_psi])
            gsm_sm[:, 1] = np.array([0, 1, 0], dtype=float)
            gsm_sm[:, 2] = np.array([-1*sin_psi, 0, cos_psi])
            self['Transform']['GSM_SM'] = gsm_sm
            self['Transform']['SM_GSM'] = gsm_sm.T
            # Get conversion to MOD for completeness
            self['Transform']['ECIMOD_SM'] = gsm_sm.dot(self['Transform']['ECIMOD_GSM'])
            self['Transform']['SM_ECIMOD'] = self['Transform']['ECIMOD_SM'].T

            # Set up transformation matrices for GEO <-> CDMAG
            # CDMAG is centered dipole geomagnetic
            #    Z: parallel to dipole axis.
            #    Y: perpendicular to both Z and rotation axis
            #    X: completes
            zhat = np.array([0, 0, 1])  # GEO Z-axis in GEO
            mag_y_vec = np.cross(zhat, dip)
            self._normVec(mag_y_vec)
            mag_x_vec = np.cross(mag_y_vec, dip)
            self._normVec(mag_x_vec)
            geo_cdmag = np.empty((3, 3))
            geo_cdmag[0, ...] = mag_x_vec
            geo_cdmag[1, ...] = mag_y_vec
            geo_cdmag[2, ...] = dip
            self['Transform']['GEO_CDMAG'] = geo_cdmag
            self['Transform']['CDMAG_GEO'] = geo_cdmag.T
            # Get conversion to MOD for completeness
            self['Transform']['ECIMOD_CDMAG'] = geo_cdmag.dot(self['Transform']['ECIMOD_GEO'])
            self['Transform']['CDMAG_ECIMOD'] = self['Transform']['ECIMOD_CDMAG'].T

        def convert(self, vec, sys_in, sys_out, defaults=None):
            """Convert an input vector between two coordinate systems

            Parameters
            ----------
            vec : array-like
                Input 3-vector (can be an array of input 3-vectors) to convert.
                E.g., for 2D input the array must be like [[x1, y1, z1], [x2, y2, z2]]
            sys_in : str
                String name for initial coordinate system. For supported systems,
                see module level documentation.
            sys_out : str
                String name for target coordinate system. For supported systems,
                see module level documentation.

            Other Parameters
            ----------------
            defaults : namedtuple or None
                Named tuple containing default settings passed from Coordinates module
            """
            # must be at least 2D for conversion methods
            gvec = np.atleast_2d(vec)
            # Needs 3xN vectors for a broadcast dot product
            trvec = gvec.T

            # Special case geodetic transforms
            to_geodetic = False
            to_rll = False
            if sys_out == 'GDZ':
                sys_out = 'GEO'
                to_geodetic = True
            elif sys_out == 'RLL':
                sys_out = 'GEO'
                to_rll = True
            if sys_in == 'GDZ':
                trvec = gdz_to_geo(vec) if defaults is None else gdz_to_geo(vec, geoid=defaults.ellipsoid)
                sys_in = 'GEO'
            elif sys_in == 'RLL':
                trvec = rll_to_geo(vec) if defaults is None else rll_to_geo(vec, geoid=defaults.ellipsoid)
                sys_in = 'GEO'

            if sys_in != sys_out:
                transform = '{0}_{1}'.format(sys_in, sys_out)
                if transform not in self['Transform']:
                    try:
                        # Construct requested transform via ECIMOD
                        trans1 = '{0}_{1}'.format(sys_in, 'ECIMOD')
                        trans2 = '{0}_{1}'.format('ECIMOD', sys_out)
                        assert trans1 in self['Transform']
                        assert trans2 in self['Transform']
                        tmatr = self['Transform'][trans2].dot(self['Transform'][trans1])
                        self['Transform'][transform] = tmatr
                    except (KeyError, AssertionError):
                        # Can't construct the transform requested
                        self._raiseErr(ValueError, 'transform')
                else:
                    # Required transform stored already, just use it
                    tmatr = self['Transform'][transform]

                converted = tmatr.dot(trvec).T
            else:
                converted = trvec
            # squeeze to fix return of 1D input vectors
            # squeeze returns either input array or a view, so this isn't wasteful
            converted_squeezed = converted.squeeze()

            if to_geodetic:
                if defaults is not None:
                    converted_squeezed = geo_to_gdz(converted_squeezed.T, geoid=defaults.ellipsoid)
                else:
                    converted_squeezed = geo_to_gdz(converted_squeezed.T)
            elif to_rll:
                if defaults is not None:
                    converted_squeezed = geo_to_rll(converted_squeezed.T, geoid=defaults.ellipsoid)
                else:
                    converted_squeezed = geo_to_rll(converted_squeezed.T)
            return converted_squeezed

        def gmst(self):
            """Calculate Greenwich Mean Sidereal Time

            Notes
            -----
            The formulation used to calculate GMST is selected using the
            status of the 'pnmodel' variable in the CTrans object attributes.
            """
            if not (self.__status['time'] or self.__status['timeInProgress']):
                self.calcTimes(from_gmst=True)
            pnmodel = self.attrs['pnmodel']
            const = self['constants']
            # Greenwich Mean Sidereal Time
            UT1_P1 = self['UT1_JC']
            coeffB = 8640184.812866
            coeffC = 0.093104
            coeffD = 6.2e-6
            if pnmodel.upper() == 'IAU82':
                # total fractional part of UT1 Julian day
                fts = const.daysec * (self['UT1_JD'] % 1 + const.j2000_jd % 1)
                gmst_rad = ((const.twopi/const.daysec) * ((-19089.45159 +
                            (coeffB + (coeffC + coeffD*UT1_P1) * UT1_P1)
                            * UT1_P1) + fts)) % const.twopi
                self['GMST'] = np.rad2deg(gmst_rad)/15
                self['GMST_rad'] = gmst_rad
            elif pnmodel.upper() == 'IAU00':
                du = self['UT1_JD'] - const.j2000_jd
                theta = const.twopi*(0.7790572732640 + 0.00273781191135448*du + du % 1) % const.twopi
                t = self['TT_JC']
                angle = (0.014506 +
                         (4612.15739966 +
                          (1.39667721 +
                           (- 0.00009344 +
                            (0.00001882)
                            * t) * t) * t) * t)
                self['GMST_rad'] = (theta + angle*const.arcsec) % const.twopi
                self['GMST'] = np.rad2deg(self['GMST_rad'])/15
            else:
                raise Exception

        def _calcSun(self):
            """
            Calculate the Sun vector.

            This should only be called from within calcCoreTransforms, after
            the ECI2000 <-> ECIMOD transforms have been defined
            """
            validEntry = self.__status['time'] and self.__status['orbital']
            validEntry = validEntry and (self.__status['coreInProgress']
                                         or self.__status['transformCore'])
            if not validEntry:
                self._raiseErr(RuntimeError, 'sun')
            const = self['constants']
            eccen = self['Eccentricity']
            cos_epsilon = np.cos(self['ObliquityEcliptic_rad'])
            sin_epsilon = np.sin(self['ObliquityEcliptic_rad'])
            cos_lambda = np.cos(self['LambdaSun'])
            sin_lambda = np.sin(self['LambdaSun'])

            if self.attrs['ephmodel'] == 'LGMDEFAULT':
                earth_sun_distance = const.AU*((1.0 - eccen * eccen)
                                               / (1.0 + eccen * np.cos(self['TrueAnomaly']))
                                               / const.Re)
                self['EarthSunDistance_Re'] = earth_sun_distance
                # Direction of the Sun in MOD coords
                sunVec = np.empty(3)
                sunVec[0] = cos_lambda
                sunVec[1] = cos_epsilon*sin_lambda
                sunVec[2] = sin_epsilon*sin_lambda
                self['SunVector_MOD'] = sunVec
                # Convert to ECI2000
                self['SunVector_J2000'] = self.convert(sunVec, 'ECIMOD', 'ECI2000')
            elif self.attrs['ephmodel'] == 'DE421':
                raise NotImplementedError
                # TODO: get Sun vector from DE421 (read HDF5 of coeffs from LGM github?)
                #       and use numpy chebyshev module for polynomials from coeffs
                # TODO: get Earth-Sun distance as magnitude of Sun vector (put in R_E)
                # TODO: normalize Sun vector (this is in ECIJ2000)

        def _raiseErr(self, errtype, code):
            """Common error raising method

            Parameters
            ----------
            errtype : Exception
                Exception to raise
            code : str
                Error category. Defines error message.
            """
            if code == 'ephmodel':
                err = 'No alternate models of ephemerides at this time'
            elif code == 'pnmodel':
                err = 'No alternate models of precession/nutation at this time'
            elif code == 'time_in':
                err = 'Input time must be single-valued. Allowed types are datetime.datetime, ' + \
                      'spacepy.time.Ticktock, or any valid input type for spacepy.time.Ticktock'
            elif code == 'eop':
                err = 'Earth Oriention Parameters not found'
            elif code == 'sun':
                err = 'Invalid entry into Sun vector routine. Not intended for direct use.'
            elif code == 'transform':
                err = 'Requested coordinate transform not defined.'
            else:
                err = 'Operation not defined'

            raise errtype('CTrans: {0}'.format(err))

        def _kepler(self, mean_anom, ecc, tol=1e-8, maxiter=100):
            """
            Solve Kepler's equation for eccentric anomaly

            Parameters
            ----------
            mean_anom : float
                Mean anomaly in radians
            ecc : float
                Eccentricity of orbit

            Returns
            -------
            ecc_anom : float
                Eccentric anomaly in radians
            """
            ecc_anom = mean_anom + ecc * np.sin(mean_anom)

            count = 0
            error = 1
            while (error > tol) and (count < maxiter):
                ecc_anom_old = ecc_anom
                ecc_anom = ecc_anom_old + (mean_anom - ecc_anom_old + ecc * np.sin(ecc_anom_old)) \
                    / (1 - ecc * np.cos(ecc_anom_old))
                count += 1
                error = np.abs(ecc_anom - ecc_anom_old)
            if count >= maxiter:
                import warnings
                warnings.warn('Estimation of eccentric anomaly failed to converge')
            return ecc_anom

    def geo_to_gdz(geovec, units='km', geoid=WGS84):
        """
        Convert geocentric geographic (cartesian GEO) to geodetic (spherical GDZ)

        Uses Heikkinen's exact solution [#Heikkinen]_, see Zhu et al. [#Zhu] for
        details.

        Parameters
        ----------
        geovec : array-like
            Nx3 array (or array-like) of geocentric geographic [x, y, z] coordinates

        Returns
        -------
        out : numpy.ndarray
            Nx3 array of geodetic altitude, latitude, and longitude

        Notes
        -----
        .. versionadded:: 0.3.0

        References
        ----------
        .. [#Heikkinen] Heikkinen, M., "Geschlossene formeln zur berechnung raumlicher geodatischer
                koordinaten aus rechtwinkligen koordinaten", Z. Vermess., vol. 107, pp. 207-211,
                1982.
        .. [#Zhu] J. Zhu, "Conversion of Earth-centered Earth-fixed coordinates to geodetic
                coordinates," in IEEE Transactions on Aerospace and Electronic Systems, vol. 30,
                no. 3, pp. 957-961, July 1994, doi: 10.1109/7.303772.
        """
        posarr = np.atleast_2d(geovec).T
        x_geo = posarr[0, :]
        y_geo = posarr[1, :]
        z_geo = posarr[2, :]
        if units == 'Re':
            # Make sure positions are in km
            rx = x_geo*geoid['A']
            ry = y_geo*geoid['A']
            rz = z_geo*geoid['A']
        else:
            rx = x_geo
            ry = y_geo
            rz = z_geo

        rad2 = rx*rx + ry*ry
        rad = np.sqrt(rad2)
        z2 = rz*rz
        # Heikkinen's method, as written in Zhu et al.
        # Each equation not explicitly in Zhu is marked with a trailing comment
        F = 54.0*geoid['B2']*z2
        G = rad2 + geoid['1mE2']*z2 - geoid['E2']*geoid['A2mB2']
        G2 = G*G  # Square G for convenience
        c = (geoid['E4']*F*rad2)/(G*G*G)
        s = np.cbrt(1 + c + np.sqrt(c*c + 2*c))
        denom_p = s + 1/s + 1  # Shorthand for term in P's denominator
        P = F/(3*denom_p*denom_p*G2)
        Q = np.sqrt(1 + 2*geoid['E4']*P)
        # The second term in r0 can evaluate as negative...
        sqrt_for_r0 = np.abs((0.5*geoid['A2'])*(1 + 1/Q) -
                             (geoid['1mE2']*P*z2)/(Q*(1 + Q)) -
                             0.5*P*rad2)
        # Back to Heikinnen's method per Zhu
        r0 = -(geoid['E2']*P*rad)/(1 + Q) + np.sqrt(sqrt_for_r0)
        brac_u = (rad - geoid['E2']*r0)  # Shorthand for term in U and V denominator
        U = np.sqrt(brac_u*brac_u + z2)
        V = np.sqrt(brac_u*brac_u + geoid['1mE2']*z2)
        z0 = (geoid['B2']*rz)/(geoid['A']*V)
        # Heikkinen's solution has a singularity at the pole, so let's trap that here
        polemask = rad <= 1e-12
        lati_gdz = np.empty_like(ry)
        if polemask.any():
            # Make sure we assign the correct pole
            zmask = rz > 0
            nhem = np.logical_and(polemask, zmask)
            shem = np.logical_and(polemask, ~zmask)
            lati_gdz[nhem] = 90
            lati_gdz[shem] = -90
        lati_gdz[~polemask] = np.rad2deg(np.arctan((rz[~polemask] +
                                                    geoid['EP2']*z0[~polemask]) /
                                                    rad[~polemask]))
                                                    # Geodetic latitude (phi in Zhu paper)
        alti_gdz = U*(1 - geoid['B2']/(geoid['A']*V))  # Geodetic altitude [km] (h in Zhu paper)
        long_gdz = np.rad2deg(np.arctan2(ry, rx))  # Geodetic longitude (same as GEO)

        out = np.c_[alti_gdz, lati_gdz, long_gdz]
        if units == 'Re':
            # Return in Re
            out[:, 0] /= geoid['A']
        return out.squeeze()

    def gdz_to_geo(gdzvec, units='km', geoid=WGS84):
        """
        Convert geodetic (GDZ) coordinates to geocentric geographic

        Parameters
        ----------
        gdzvec : array-like
            Nx3 array of geodetic altitude, latitude, longitude (in specified units)

        Returns
        -------
        out : numpy.ndarray
            Nx3 array of geocentric geographic x, y, z coordinates

        Other Parameters
        ----------------
        units : str
            Units for input geodetic altitude. Options are 'km' or 'Re'. Default is 'km'.
            Output units will be the same as input units.
        geoid : spacepy.ctrans.Ellipsoid
            Instance of a reference ellipsoid to use for geodetic conversion.
            Default is WGS84.

        Notes
        -----
        .. versionadded:: 0.3.0
        """
        posarr = np.atleast_2d(gdzvec)
        h = posarr[:, 0] if units == 'km' else posarr[:, 0]*geoid['A']  # convert to km
        lam = np.deg2rad(posarr[:, 1])
        phi = np.deg2rad(posarr[:, 2])
        chi = np.sqrt(1 - geoid['E2']*np.sin(lam)*np.sin(lam))

        # Convert to GEO [km]
        x = (geoid['A']/chi + h)*np.cos(lam)*np.cos(phi)
        y = (geoid['A']/chi + h)*np.cos(lam)*np.sin(phi)
        z = (geoid['A']*(1 - geoid['E2'])/chi + h)*np.sin(lam)

        out = np.c_[x, y, z]
        if units == 'km':
            return out.squeeze()
        else:
            # Return in Re
            return out.squeeze()/geoid['A']

    def geo_to_rll(geovec, units='km', geoid=WGS84):
        """Calculate RLL from geocentric geographic (GEO) coordinates

        Parameters
        ----------
        geovec : array-like
            Nx3 array of geographic radius, latitude, longitude (in specified units)

        Returns
        -------
        rllvec : numpy.ndarray
            Nx3 array of [distance from Earth's center, geodetic latitude, geodetic longitude]

        Other Parameters
        ----------------
        units : str
            Units for input geodetic altitude. Options are 'km' or 'Re'. Default is 'km'.
            Output units will be the same as input units.
        geoid : spacepy.ctrans.Ellipsoid
            Instance of a reference ellipsoid to use for geodetic conversion.
            Default is WGS84.

        Notes
        -----
        .. versionadded:: 0.3.0
        """
        rllvec = np.atleast_2d(geo_to_gdz(geovec, units=units, geoid=geoid))
        # Reolace altitude with norm of Cartesian position
        rllvec[:, 0] = np.linalg.norm(geovec, axis=-1)

        return rllvec.squeeze()

    def rll_to_geo(rllvec, units='km', geoid=WGS84):
        """Calculate geocentric geographic (GEO) from RLL coordinates

        Parameters
        ----------
        rllvec : array-like
            Nx3 array of geocentric radius, geodetic latitude, geodetic longitude
            (in specified units)

        Returns
        -------
        geoarr : numpy.ndarray
            Nx3 array of [altitude, geodetic latitude, geodetic longitude]

        Other Parameters
        ----------------
        units : str
            Units for input geocentric radii. Options are 'km' or 'Re'. Default is 'km'.
            Output units will be the same as input units.
        geoid : spacepy.ctrans.Ellipsoid
            Instance of a reference ellipsoid to use for geodetic conversion.
            Default is WGS84.

        Notes
        -----
        .. versionadded:: 0.3.0
        """
        posarr = np.atleast_2d(rllvec).astype(float)
        surf = np.zeros_like(posarr)
        surf[:, 1:] = posarr[:, 1:]
        geoid_at_pos = np.atleast_2d(gdz_to_geo(surf, units=units, geoid=geoid))
        gdz = posarr.copy()
        # remove geoid height from geocentric radius at each location
        gdz[:, 0] -= np.linalg.norm(geoid_at_pos, axis=-1)
        geoarr = gdz_to_geo(gdz, units=units, geoid=geoid)

        return geoarr

    def convert_multitime(coords, ticks, sys_in, sys_out, defaults=None, itol=None):
        '''
        Convert coordinates for N times, where N >= 1

        Parameters
        ----------
        coords : array-like
            Coordinates as Nx3 array. Cartesian assumed unless input system is geodetic.
        ticks : spacepy.time.Ticktock
            Times for each element of coords. Must contain either N times or 1 time.
        sys_in : str
            Name of input coordinate system.
        sys_out : str
            Name of output coordinate system.

        Other Parameters
        ----------------
        defaults : namedtuple or None
            Named tuple with parameters from coordinates module
        itol : float
            Time tolerance, in seconds, for using a unique set of conversions.
            Default is 1. Supplying a defaults namedtuple (i.e., if routine is called
            by spacepy.cooordinates.Coords.convert) will override this value.
        '''
        itol = itol if defaults is None else defaults.itol
        # Calculate a CTrans for each unique time
        ctdict = dict()
        ttused = set()
        tais = ticks.TAI
        for idx, tt in enumerate(tais):
            if tt not in ctdict:
                # for speed, let's not recalculate transforms
                # for times very close to each other (defined
                # by the itol kwarg)
                if ttused:
                    usedlist = list(ttused)
                    closest = np.argmin(np.abs(tt - usedlist))
                    tdiff = np.abs(tt - usedlist[closest])
                else:
                    tdiff = itol + 1
                if tdiff < itol and ctdict:
                    ctdict[tt] = ctdict[usedlist[closest]]
                else:
                    ctdict[tt] = CTrans(ticks[idx])
                    ctdict[tt].calcCoreTransforms()
                    if sys_in in magsys or sys_out in magsys:
                        ctdict[tt].calcMagTransforms()
                    ttused.add(tt)
        # Unless speed becomes an issue, let's just loop over each value
        # except the spacial case of only 1 unique time
        loopover = np.atleast_2d(coords)
        if len(ctdict) == 1:
            newcoords = ctdict[tais[0]].convert(loopover, sys_in, sys_out, defaults=defaults)
        else:
            newcoords = np.empty_like(loopover)
            for idx, cc in enumerate(loopover):
                newcoords[idx] = ctdict[tais[idx]].convert(cc, sys_in, sys_out, defaults=defaults)
        return newcoords.squeeze()