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# Source code for astropy.coordinates.angles

    # -*- coding: utf-8 -*-
    # Licensed under a 3-clause BSD style license - see LICENSE.rst

    """
    This module contains the fundamental classes used for representing
    coordinates in astropy.
    """
    from __future__ import (absolute_import, division, print_function,
                            unicode_literals)

    import math

    import numpy as np

    from ..extern import six
    from . import angle_utilities as util
    from .. import units as u
    from ..utils import deprecated

    __all__ = ['Angle', 'Latitude', 'Longitude']

    TWOPI = math.pi * 2.0  # no need to calculate this all the time

    class Angle(u.Quantity):
        """
        One or more angular value(s) with units equivalent to radians or degrees.

        An angle can be specified either as an array, scalar, tuple (see
        below), string, `~astropy.units.Quantity` or another
        `~astropy.coordinates.Angle`.

        The input parser is flexible and supports a variety of formats::

          Angle('10.2345d')
          Angle(['10.2345d', '-20d'])
          Angle('1:2:30.43 degrees')
          Angle('1 2 0 hours')
          Angle(np.arange(1, 8), unit=u.deg)
          Angle(u'1°2′3″')
          Angle('1d2m3.4s')
          Angle('-1h2m3s')
          Angle((-1, 2, 3), unit=u.deg)  # (d, m, s)
          Angle(10.2345 * u.deg)
          Angle(Angle(10.2345 * u.deg))

        Parameters
        ----------
        angle : array, scalar, Quantity, Angle
            The angle value. If a tuple, will be interpreted as ``(h, m
            s)`` or ``(d, m, s)`` depending on `unit`. If a string, it
            will be interpreted following the rules described above.

            If `angle` is a sequence or array of strings, the resulting
            values will be in the given `unit`, or if None is provided,
            the unit will be taken from the first given value.

        unit : `~astropy.units.UnitBase`, str, optional
            The unit of the value specified for the angle.  This may be
            any string that `~astropy.units.Unit` understands, but it is
            better to give an actual unit object.  Must be an angular
            unit.

        dtype : ~numpy.dtype, optional
            See `~astropy.units.Quantity`.

        copy : bool, optional
            See `~astropy.units.Quantity`.

        Raises
        ------
        `~astropy.units.core.UnitsError`
            If a unit is not provided or it is not an angular unit.
        """
        _include_easy_conversion_members = True

        def __new__(cls, angle, unit=None, dtype=None, copy=True):
            unit = cls._convert_unit_to_angle_unit(unit)
            if (unit is not None and
                not unit.is_equivalent(u.radian)):
                raise u.UnitsError(
                    "Given unit {0} is not convertible to an angle".format(
                        unit))

            if isinstance(angle, u.Quantity):
                # This includes Angle subclasses as well
                if unit is not None:
                    angle = angle.to(unit).value
                else:
                    unit = angle.unit
                    unit = cls._convert_unit_to_angle_unit(unit)
                    if not unit.is_equivalent(u.radian):
                        raise u.UnitsError(
                            "Given quantity {0} is not convertible to an angle".format(
                                angle))

                    angle = angle.value

            angle = cls._tuple_to_float(angle, unit)

            try:
                angle = np.asarray(angle)
            except ValueError as e:
                raise TypeError(str(e))

            if angle.dtype.type in (np.string_, np.unicode_):
                # We need to modify this value from within
                # convert_string_to_angle, and the only way to do that
                # across Python 2.6 - 3.3 is to use this "store it in a
                # list" trick.
                determined_unit = [unit]

                def convert_string_to_angle(x):
                    ang, new_unit = util.parse_angle(six.text_type(x), unit)
                    if determined_unit[0] is None:
                        determined_unit[0] = new_unit
                    if new_unit is not None:
                        return new_unit.to(
                            determined_unit[0], cls._tuple_to_float(ang, new_unit))
                    else:
                        return cls._tuple_to_float(ang, determined_unit[0])

                convert_string_to_angle_ufunc = np.vectorize(
                    convert_string_to_angle,
                    otypes=[np.float_])
                angle = convert_string_to_angle_ufunc(angle)
                unit = determined_unit[0]

            elif angle.dtype.kind not in 'iuf':
                raise TypeError("Unsupported dtype '{0}'".format(angle.dtype))

            if unit is None:
                raise u.UnitsError("No unit was specified")

            self = super(Angle, cls).__new__(
                cls, angle, unit, dtype=dtype, copy=copy)

            return self

        @staticmethod
        def _tuple_to_float(angle, unit):
            """
            Converts an angle represented as a 3-tuple into a floating
            point number in the given unit.
            """
            if isinstance(angle, tuple):
                # TODO: Numpy array of tuples?
                if unit is u.hourangle:
                    util.check_hms_ranges(*angle)
                    angle = util.hms_to_hours(*angle)
                elif unit is u.degree:
                    angle = util.dms_to_degrees(*angle)
                else:
                    raise u.UnitsError(
                        "Can not parse '{0}' as unit '{1}'".format(
                            angle, unit))
            return angle

        @staticmethod
        def _convert_unit_to_angle_unit(unit):
            if unit is not None:
                unit = u.Unit(unit)

                if unit is u.hour:
                    unit = u.hourangle
            return unit

        def __quantity_view__(self, obj, unit):
            unit = self._convert_unit_to_angle_unit(unit)
            if unit is not None and unit.is_equivalent(u.radian):
                result = obj.view(Angle)
                return result
            return super(Angle, self).__quantity_view__(
                obj, unit)

        def __quantity_instance__(self, val, unit, **kwargs):
            unit = self._convert_unit_to_angle_unit(unit)
            if unit is not None and unit.is_equivalent(u.radian):
                return self.__class__(val, unit, **kwargs)
            return super(Angle, self).__quantity_instance__(val, unit, **kwargs)

        def __array_wrap__(self, obj, context=None):
            obj = super(Angle, self).__array_wrap__(obj, context=context)

            if isinstance(obj, Angle):
                return Angle(obj.value, obj.unit)

            return obj

        def __add__(self, other):
            return super(Angle, self).__add__(other)

        def __sub__(self, other):
            return super(Angle, self).__sub__(other)

        def __mul__(self, other):
            if isinstance(other, type(self)):
                raise TypeError(
                    "multiplication is not supported between two {0} "
                    "objects".format(
                        type(self).__name__))
            return super(Angle, self).__mul__(other)

        def __div__(self, other):
            if isinstance(other, type(self)):
                raise TypeError(
                    "division is not supported between two {0} objects".format(
                        type(self).__name__))
            return super(Angle, self).__div__(other)

        __truediv__ = __div__

        @property
        def hour(self):
            """
            The angle's value in hours (read-only property).
            """
            return self.hourangle

        @property
        def hms(self):
            """
            The angle's value in hours, as a ``(h, m, s)`` tuple
            (read-only property).
            """
            return util.hours_to_hms(self.hourangle)

        @property
        def dms(self):
            """
            The angle's value in degrees, as a ``(d, m, s)`` tuple
            (read-only property).
            """
            return util.degrees_to_dms(self.degree)

        def to_string(self, unit=None, decimal=False, sep='fromunit',
                      precision=None, alwayssign=False, pad=False,
                      fields=3, format=None):
            """ A string representation of the angle.

            Parameters
            ----------
            units : `~astropy.units.UnitBase`, optional
                Specifies the units.  Must be an angular unit.  If not
                provided, the unit used to initialize the angle will be
                used.

            decimal : bool, optional
                If True, a decimal respresentation will be used, otherwise
                the returned string will be in sexagesimal form.

            sep : str, optional
                The separator between numbers in a sexagesimal
                representation.  E.g., if it is ':', the result is
                "12:41:11.1241". Also accepts 2 or 3 separators. E.g.,
                ``sep='hms'`` would give the result "12h41m11.1241s", or
                sep='-:' would yield "11-21:17.124".  Alternatively, the
                special string 'fromunit' means 'dms' if the unit is
                degrees, or 'hms' if the unit is hours.

            precision : int, optional
                The level of decimal precision.  If `decimal` is True,
                this is the raw precision, otherwise it gives the
                precision of the last place of the sexagesimal
                representation (seconds).  If `None`, or not provided, the
                number of decimal places is determined by the value, and
                will be between 0-8 decimal places as required.

            alwayssign : bool, optional
                If `True`, include the sign no matter what.  If `False`,
                only include the sign if it is negative.

            pad : bool, optional
                If `True`, include leading zeros when needed to ensure a
                fixed number of characters for sexagesimal representation.

            fields : int, optional
                Specifies the number of fields to display when outputting
                sexagesimal notation.  For example:

                    - fields == 1: `'5d'`
                    - fields == 2: `'5d45m'`
                    - fields == 3: `'5d45m32.5s'`

                By default, all fields are displayed.

            format : str, optional
                The format of the result.  If not provided, an unadorned
                string is returned.  Supported values are:

                - 'latex': Return a LaTeX-formatted string

                - 'unicode': Return a string containing non-ASCII unicode
                  characters, such as the degree symbol

            Returns
            -------
            strrepr : str
                A string representation of the angle.

            """
            if unit is None:
                unit = self.unit
            unit = self._convert_unit_to_angle_unit(unit)

            separators = {
                None: {
                    u.degree: 'dms',
                    u.hourangle: 'hms'},
                'latex': {
                    u.degree: [r'^\circ', r'{}^\prime', r'{}^{\prime\prime}'],
                    u.hourangle: [r'^\mathrm{h}', r'^\mathrm{m}', r'^\mathrm{s}']},
                'unicode': {
                    u.degree: '°′″',
                    u.hourangle: 'ʰᵐˢ'}
                }

            if sep == 'fromunit':
                if format not in separators:
                    raise ValueError("Unknown format '{0}'".format(format))
                seps = separators[format]
                if unit in seps:
                    sep = seps[unit]

            # Create an iterator so we can format each element of what
            # might be an array.
            if unit is u.degree:
                if decimal:
                    values = self.degree
                    if precision is not None:
                        func = ("{0:0." + str(precision) + "f}").format
                    else:
                        func = '{0:g}'.format
                else:
                    if sep == 'fromunit':
                        sep = 'dms'
                    values = self.degree
                    func = lambda x: util.degrees_to_string(
                        x, precision=precision, sep=sep, pad=pad,
                        fields=fields)

            elif unit is u.hourangle:
                if decimal:
                    values = self.hour
                    if precision is not None:
                        func = ("{0:0." + str(precision) + "f}").format
                    else:
                        func = '{0:g}'.format
                else:
                    if sep == 'fromunit':
                        sep = 'hms'
                    values = self.hour
                    func = lambda x: util.hours_to_string(
                        x, precision=precision, sep=sep, pad=pad,
                        fields=fields)

            elif unit.is_equivalent(u.radian):
                if decimal:
                    values = self.to(unit).value
                    if precision is not None:
                        func = ("{0:1." + str(precision) + "f}").format
                    else:
                        func = "{0:g}".format
                elif sep == 'fromunit':
                    values = self.to(unit).value
                    unit_string = unit.to_string(format=format)
                    if format == 'latex':
                        unit_string = unit_string[1:-1]

                    if precision is not None:
                        def plain_unit_format(val):
                            return ("{0:0." + str(precision) + "f}{1}").format(
                                val, unit_string)
                        func = plain_unit_format
                    else:
                        def plain_unit_format(val):
                            return "{0:g}{1}".format(val, unit_string)
                        func = plain_unit_format
                else:
                    raise ValueError(
                        "'{0}' can not be represented in sexagesimal "
                        "notation".format(
                            unit.name))

            else:
                raise u.UnitsError(
                    "The unit value provided is not an angular unit.")

            def do_format(val):
                s = func(float(val))
                if alwayssign and not s.startswith('-'):
                    s = '+' + s
                if format == 'latex':
                    s = '${0}$'.format(s)
                return s

            format_ufunc = np.vectorize(do_format, otypes=[np.object])
            result = format_ufunc(values)
            if result.ndim == 0:
                result = result[()]
            return result

        def wrap_at(self, wrap_angle, inplace=False):
            """
            Wrap the Angle object at the given ``wrap_angle``.

            This method forces all the angle values to be within a contiguous 360 degree
            range so that ``wrap_angle - 360d <= angle < wrap_angle``.  By default a new
            Angle object is returned, but if the ``inplace`` argument is ``True`` then
            the Angle object is wrapped in place and nothing is returned.

            For instance::

              >>> from astropy.coordinates import Angle
              >>> import astropy.units as u
              >>> a = Angle([-20.0, 150.0, 350.0] * u.deg)

              >>> a.wrap_at(360 * u.deg).degree  # Wrap into range 0 to 360 degrees
              array([ 340.,  150.,  350.])

              >>> a.wrap_at('180d', inplace=True)  # Wrap into range -180 to 180 degrees
              >>> a.degree
              array([ -20.,  150.,  -10.])

            Parameters
            ----------
            wrap_angle : str, Angle, angular Quantity
                Specifies a single value for the wrap angle.  This can be any
                object that can initialize an Angle object, e.g. '180d', 180 * u.deg,
                or Angle(180, unit=u.deg).

            inplace : bool
                If ``True`` then wrap the object in place instead of returning a new Angle

            Returns
            -------
            out : Angle or None
                If ``inplace is False`` (default), return new Angle object with angles
                wrapped accordingly.  Otherwise wrap in place and return None.
            """
            wrap_angle = Angle(wrap_angle)  # Convert to an Angle
            wrapped = np.mod(self - wrap_angle, 360.0 * u.deg) - (360.0 * u.deg - wrap_angle)

            if inplace:
                self[()] = wrapped
            else:
                return wrapped

        def is_within_bounds(self, lower=None, upper=None):
            """
            Check if all angle(s) satisfy ``lower <= angle < upper``

            If ``lower`` is not specified (or ``None``) then no lower bounds check is
            performed.  Likewise ``upper`` can be left unspecified.  For example::

              >>> from astropy.coordinates import Angle
              >>> import astropy.units as u
              >>> a = Angle([-20, 150, 350] * u.deg)
              >>> a.is_within_bounds('0d', '360d')
              False
              >>> a.is_within_bounds(None, '360d')
              True
              >>> a.is_within_bounds(-30 * u.deg, None)
              True

            Parameters
            ----------
            lower : str, Angle, angular Quantity, None
                Specifies lower bound for checking.  This can be any object
                that can initialize an Angle object, e.g. '180d', 180 * u.deg,
                or Angle(180, unit=u.deg).
            upper : str, Angle, angular Quantity, None
                Specifies upper bound for checking.  This can be any object
                that can initialize an Angle object, e.g. '180d', 180 * u.deg,
                or Angle(180, unit=u.deg).

            Returns
            -------
            is_within_bounds : bool
                True if all angles satisfy ``lower <= angle < upper``
            """
            ok = True
            if lower is not None:
                ok &= np.all(Angle(lower) <= self)
            if ok and upper is not None:
                ok &= np.all(self < Angle(upper))
            return bool(ok)

        @deprecated("0.3", name="format", alternative="to_string")
        def format(self, unit=u.degree, decimal=False, sep='fromunit', precision=5,
                   alwayssign=False, pad=False):
            return self.to_string(
                unit=unit, decimal=decimal, sep=sep, precision=precision,
                alwayssign=alwayssign, pad=pad)

        def __str__(self):
            return str(self.to_string())

        def _repr_latex_(self):
            return str(self.to_string(format='latex'))

    class Latitude(Angle):
        """
        Latitude-like angle(s) which must be in the range -90 to +90 deg.

        A Latitude object is distinguished from a pure `Angle` by virtue
        of being constrained so that::

          -90.0 * u.deg <= angle(s) <= +90.0 * u.deg

        Any attempt to set a value outside that range will result in a `ValueError`.

        The input angle(s) can be specified either as an array, list, scalar, tuple (see
        below), string, :class:`~astropy.units.quantity.Quantity` or another `Angle`.

        The input parser is flexible and supports all of the input formats supported by `Angle`.

        Parameters
        ----------
        angle : array, list, scalar, Quantity, Angle
            The angle value(s). If a tuple, will be interpreted as ``(h, m
            s)`` or ``(d, m, s)`` depending on `unit`. If a string, it
            will be interpreted following the rules described for `Angle`.

            If `angle` is a sequence or array of strings, the resulting
            values will be in the given `unit`, or if None is provided,
            the unit will be taken from the first given value.

        unit : :class:`~astropy.units.core.UnitBase`, str, optional
            The unit of the value specified for the angle.  This may be
            any string that `~astropy.units.Unit` understands, but it is
            better to give an actual unit object.  Must be an angular
            unit.

        Raises
        ------
        `~astropy.units.core.UnitsError`
            If a unit is not provided or it is not an angular unit.
        """
        def __new__(cls, angle, unit=None, **kwargs):
            self = super(Latitude, cls).__new__(cls, angle, unit=unit, **kwargs)
            self._validate_angles()
            return self

        def _validate_angles(self):
            if np.any(self < -90.0 * u.deg) or np.any(self > 90.0 * u.deg):
                raise ValueError('Latitude angle(s) must be within -90 deg <= angle <= 90 deg, '
                                 'got {0}'.format(self.degree))

        def __setitem__(self, item, value):
            super(Latitude, self).__setitem__(item, value)
            self._validate_angles()

    class Longitude(Angle):
        """
        Longitude-like angle(s) which are wrapped within a contiguous 360 degree range.

        A ``Longitude`` object is distinguished from a pure `~astropy.coordinates.Angle` by virtue
        of a ``wrap_angle`` property.  The ``wrap_angle`` specifies that all angle values
        represented by the object will be in the range::

          wrap_angle - 360 * u.deg <= angle(s) < wrap_angle

        The default ``wrap_angle`` is 360 deg.  Setting ``wrap_angle=180 * u.deg`` would
        instead result in values between -180 and +180 deg.  Setting the ``wrap_angle``
        attribute of an existing ``Longitude`` object will result in re-wrapping the
        angle values in-place.

        The input angle(s) can be specified either as an array, list, scalar, tuple,
        string, :class:`~astropy.units.quantity.Quantity` or another `Angle`.

        The input parser is flexible and supports all of the input formats supported by `Angle`.

        Parameters
        ----------
        angle : array, list, scalar, Quantity, `Angle`
            The angle value(s). If a tuple, will be interpreted as ``(h, m
            s)`` or ``(d, m, s)`` depending on `unit`. If a string, it
            will be interpreted following the rules described for `Angle`.

            If `angle` is a sequence or array of strings, the resulting
            values will be in the given `unit`, or if None is provided,
            the unit will be taken from the first given value.

        unit : :class:`~astropy.units.core.UnitBase`, str, optional
            The unit of the value specified for the angle.  This may be
            any string that `~astropy.units.Unit` understands, but it is
            better to give an actual unit object.  Must be an angular
            unit.

        wrap_angle : `Angle` or equivalent
            Angle at which to wrap back to ``wrap_angle - 360 deg``.

        Raises
        ------
        `~astropy.units.core.UnitsError`
            If a unit is not provided or it is not an angular unit.
        """
        def __new__(cls, angle, unit=None, wrap_angle=360 * u.deg, **kwargs):
            self = super(Longitude, cls).__new__(cls, angle, unit=unit, **kwargs)
            self.wrap_angle = wrap_angle
            return self

        def __setitem__(self, item, value):
            super(Longitude, self).__setitem__(item, value)
            self._wrap_internal()

        def _wrap_internal(self):
            """
            Wrap the internal values in the Longitude object.  Using the `Angle`
            wrap_at() method causes recursion.
            """
            d360 = 360.0 * u.deg
            wrapped = np.mod(self - self.wrap_angle, d360) - (d360 - self.wrap_angle)
            super(Longitude, self).__setitem__((), wrapped)

        @property
        def wrap_angle(self):
            return self._wrap_angle

        @wrap_angle.setter
        def wrap_angle(self, value):
            self._wrap_angle = Angle(value)
            self._wrap_internal()

        def __quantity_view__(self, obj, unit):
            unit = self._convert_unit_to_angle_unit(unit)
            if unit is not None and unit.is_equivalent(u.radian):
                # by default, wrap_angle and equivalencies remain the same
                # TODO: generalize to some _things_to_copy once #1422, #1373 merged
                return obj.view(Longitude)
            return super(Angle, self).__quantity_view__(obj, unit)

        def __quantity_instance__(self, val, unit, **kwargs):
            unit = self._convert_unit_to_angle_unit(unit)
            if unit is not None and unit.is_equivalent(u.radian):
                # by default, wrap_angle remains the same
                if 'wrap_angle' not in kwargs:
                    kwargs['wrap_angle'] = getattr(self, 'wrap_angle')
                return Longitude(val, unit, **kwargs)
            return super(Angle, self).__quantity_instance__(val, unit, **kwargs)

        def __getitem__(self, key):
            out = super(Longitude, self).__getitem__(key)
            out._wrap_angle = self._wrap_angle
            return out

        # deprecated; TODO: move to quantity later (once #1422, #1373 merged)
        def __getslice__(self, i, j):
            return self.__getitem__(slice(i, j))

    #<----------------------------------Rotations--------------------------------->

    def rotation_matrix(angle, axis='z', unit=None):
        """
        Generate a 3x3 cartesian rotation matrix in for rotation about
        a particular axis.

        Parameters
        ----------
        angle : convertible to Angle
            The amount of rotation this matrix should represent.

        axis : str or 3-sequence
            Either 'x','y', 'z', or a (x,y,z) specifying an axis to rotate
            about. If 'x','y', or 'z', the rotation sense is
            counterclockwise looking down the + axis (e.g. positive
            rotations obey left-hand-rule).

        unit : UnitBase, optional
            If `angle` does not have associated units, they are in this
            unit.  If neither are provided, it is assumed to be degrees.

        Returns
        -------
        rmat: `numpy.matrix`
            A unitary rotation matrix.
        """
        # TODO: This doesn't handle arrays of angles

        if unit is None:
            unit = u.degree

        angle = Angle(angle, unit=unit)

        if axis == 'z':
            s = np.sin(angle)
            c = np.cos(angle)
            return np.matrix(((c, s, 0),
                              (-s, c, 0),
                              (0, 0, 1)))
        elif axis == 'y':
            s = np.sin(angle)
            c = np.cos(angle)
            return np.matrix(((c, 0, -s),
                              (0, 1, 0),
                              (s, 0, c)))
        elif axis == 'x':
            s = np.sin(angle)
            c = np.cos(angle)
            return np.matrix(((1, 0, 0),
                              (0, c, s),
                              (0, -s, c)))
        else:
            x, y, z = axis
            w = np.cos(angle / 2)

            # normalize
            if w == 1:
                x = y = z = 0
            else:
                l = np.sqrt((x * x + y * y + z * z) / (1 - w * w))
                x /= l
                y /= l
                z /= l

            wsq = w * w
            xsq = x * x
            ysq = y * y
            zsq = z * z
            return np.matrix(((wsq + xsq - ysq - zsq, 2 * x * y - 2 * w * z, 2 * x * z + 2 * w * y),
                              (2 * x * y + 2 * w * z, wsq - xsq + ysq - zsq, 2 * y * z - 2 * w * x),
                              (2 * x * z - 2 * w * y, 2 * y * z + 2 * w * x, wsq - xsq - ysq + zsq)))

    def angle_axis(matrix, unit=None):
        """
        Computes the angle of rotation and the rotation axis for a given rotation
        matrix.

        Parameters
        ----------
        matrix : array-like
            A 3 x 3 unitary rotation matrix.

        unit : UnitBase
            The output unit.  If `None`, the output unit is degrees.

        Returns
        -------
        angle : Angle
            The angle of rotation for this matrix.

        axis : array (length 3)
            The axis of rotation for this matrix.
        """
        # TODO: This doesn't handle arrays of angles

        m = np.asmatrix(matrix)
        if m.shape != (3, 3):
            raise ValueError('matrix is not 3x3')

        angle = np.acos((m[0, 0] + m[1, 1] + m[2, 2] - 1) / 2)
        denom = np.sqrt(2 * ((m[2, 1] - m[1, 2]) + (m[0, 2] - m[2, 0]) + (m[1, 0] - m[0, 1])))
        axis = np.array((m[2, 1] - m[1, 2], m[0, 2] - m[2, 0], m[1, 0] - m[0, 1])) / denom
        axis /= np.sqrt(np.sum(axis ** 2))

        angle = Angle(angle, u.radian)
        if unit is None:
            unit = u.degree
        return angle.to(unit), axis