.. _working_with_angles:

Working with Angles
*******************

The angular components of the various coordinate objects are represented
by objects of the |Angle| class. While most likely to be encountered in
the context of coordinate objects, |Angle| objects can also be used on
their own wherever a representation of an angle is needed.

.. _angle-creation:

Creation
========

The creation of an |Angle| object is quite flexible and supports a wide
variety of input object types and formats. The type of the input angle(s)
can be array, scalar, tuple, string, `~astropy.units.Quantity` or another
|Angle|. This is best illustrated with a number of examples of valid ways
to create an |Angle|.

Examples
--------

..
  EXAMPLE START
  Different Ways to Create an Angle Object

There are a number of ways to create an |Angle|::

    >>> import numpy as np
    >>> from astropy import units as u
    >>> from astropy.coordinates import Angle

    >>> Angle('10.2345d')              # String with 'd' abbreviation for degrees  # doctest: +FLOAT_CMP
    <Angle 10.2345 deg>
    >>> Angle(['10.2345d', '-20d'])    # Array of strings  # doctest: +FLOAT_CMP
    <Angle [ 10.2345, -20.    ] deg>
    >>> Angle('1:2:30.43 degrees')     # Sexagesimal degrees  # doctest: +FLOAT_CMP
    <Angle 1.04178611 deg>
    >>> Angle('1 2 0 hours')           # Sexagesimal hours  # doctest: +FLOAT_CMP
    <Angle 1.03333333 hourangle>
    >>> Angle(np.arange(1., 8.), unit=u.deg)  # Numpy array from 1..7 in degrees  # doctest: +FLOAT_CMP
    <Angle [1., 2., 3., 4., 5., 6., 7.] deg>
    >>> Angle('1°2′3″')               # Unicode degree, arcmin and arcsec symbols  # doctest: +FLOAT_CMP
    <Angle 1.03416667 deg>
    >>> Angle('1°2′3″N')               # Unicode degree, arcmin, arcsec symbols and direction  # doctest: +FLOAT_CMP
    <Angle 1.03416667 deg>
    >>> Angle('1d2m3.4s')              # Degree, arcmin, arcsec.  # doctest: +FLOAT_CMP
    <Angle 1.03427778 deg>
    >>> Angle('1d2m3.4sS')              # Degree, arcmin, arcsec, direction.  # doctest: +FLOAT_CMP
    <Angle -1.03427778 deg>
    >>> Angle('-1h2m3s')               # Hour, minute, second  # doctest: +FLOAT_CMP
    <Angle -1.03416667 hourangle>
    >>> Angle('-1h2m3sW')               # Hour, minute, second, direction  # doctest: +FLOAT_CMP
    <Angle 1.03416667 hourangle>
    >>> Angle(10.2345 * u.deg)         # From a Quantity object in degrees  # doctest: +FLOAT_CMP
    <Angle 10.2345 deg>
    >>> Angle(Angle(10.2345 * u.deg))  # From another Angle object  # doctest: +FLOAT_CMP
    <Angle 10.2345 deg>

..
  EXAMPLE END

Representation
==============

The |Angle| object also supports a variety of ways of representing the value
of the angle, both as a floating point number and as a string.

Examples
--------

..
  EXAMPLE START
  Representation of Angle Object Values

There are many ways to represent the value of an |Angle|::

    >>> a = Angle(1, u.radian)
    >>> a  # doctest: +FLOAT_CMP
    <Angle 1. rad>
    >>> a.radian
    1.0
    >>> a.degree  # doctest: +FLOAT_CMP
    57.29577951308232
    >>> a.hour  # doctest: +FLOAT_CMP
    3.8197186342054885
    >>> a.hms  # doctest: +FLOAT_CMP
    hms_tuple(h=3.0, m=49.0, s=10.987083139758766)
    >>> a.dms  # doctest: +FLOAT_CMP
    dms_tuple(d=57.0, m=17.0, s=44.806247096362313)
    >>> a.signed_dms  # doctest: +FLOAT_CMP
    signed_dms_tuple(sign=1.0, d=57.0, m=17.0, s=44.806247096362313)
    >>> (-a).dms  # doctest: +FLOAT_CMP
    dms_tuple(d=-57.0, m=-17.0, s=-44.806247096362313)
    >>> (-a).signed_dms  # doctest: +FLOAT_CMP
    signed_dms_tuple(sign=-1.0, d=57.0, m=17.0, s=44.806247096362313)
    >>> a.arcminute  # doctest: +FLOAT_CMP
    3437.7467707849396
    >>> f"{a}"
    '1.0 rad'
    >>> f"{a:latex}"
    '$1\\mathrm{rad}$'
    >>> f"{a.to(u.deg):latex}"
    '$57^\\circ17{}^\\prime44.8062471{}^{\\prime\\prime}$'
    >>> a.to_string()
    '1rad'
    >>> a.to_string(unit=u.degree)
    '57d17m44.8062471s'
    >>> a.to_string(unit=u.degree, sep=':')
    '57:17:44.8062471'
    >>> a.to_string(unit=u.degree, sep=('deg', 'm', 's'))
    '57deg17m44.8062471s'
    >>> a.to_string(unit=u.hour)
    '3h49m10.98708314s'
    >>> a.to_string(unit=u.hour, decimal=True)
    '3.81972'

..
  EXAMPLE END

Usage
=====

Angles will also behave correctly for appropriate arithmetic operations.

Example
-------

..
  EXAMPLE START
  Arithmetic Operations Using Angle Objects

To use |Angle| objects in arithmetic operations::

    >>> a = Angle(1.0, u.radian)
    >>> a + 0.5 * u.radian + 2 * a  # doctest: +FLOAT_CMP
    <Angle 3.5 rad>
    >>> np.sin(a / 2)  # doctest: +FLOAT_CMP
    <Quantity 0.47942554>
    >>> a == a  # doctest: +SKIP
    array(True, dtype=bool)
    >>> a == (a + a)    # doctest: +SKIP
    array(False, dtype=bool)

..
  EXAMPLE END

|Angle| objects can also be used for creating coordinate objects.

Example
-------

..
  EXAMPLE START
  Creating Coordinate Objects with Angle Objects

To create a coordinate object using an |Angle|::

    >>> from astropy.coordinates import ICRS
    >>> ICRS(Angle(1, u.deg), Angle(0.5, u.deg))  # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec) in deg
        (1., 0.5)>

..
  EXAMPLE END

Wrapping and Bounds
===================

There are two utility methods for working with angles that should have bounds.
The :meth:`~astropy.coordinates.Angle.wrap_at` method allows taking an angle or
angles and wrapping to be within a single 360 degree slice. The
:meth:`~astropy.coordinates.Angle.is_within_bounds` method returns a
boolean indicating whether an angle or angles is within the specified bounds.


Longitude and Latitude Objects
==============================

|Longitude| and |Latitude| are two specialized subclasses of the |Angle|
class that are used for all of the spherical coordinate classes.
|Longitude| is used to represent values like right ascension, Galactic
longitude, and azimuth (for Equatorial, Galactic, and Alt-Az coordinates,
respectively). |Latitude| is used for declination, Galactic latitude, and
elevation.

Longitude
---------

A |Longitude| object is distinguished from a pure |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. For example::

    >>> from astropy.coordinates import Longitude
    >>> a = Longitude([-20, 150, 350, 360] * u.deg)
    >>> a.degree  # doctest: +FLOAT_CMP
    array([340., 150., 350.,   0.])
    >>> a.wrap_angle = 180 * u.deg
    >>> a.degree  # doctest: +FLOAT_CMP
    array([-20., 150., -10.,   0.])

Latitude
--------

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

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

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


Generating Angle Values
=======================

Astropy provides utility functions for generating angular or spherical
positions, either with random sampling or with a grid of values. These functions
all return `~astropy.coordinates.BaseRepresentation` subclass instances, which
can be passed directly into coordinate frame classes or |SkyCoord| to create
random or gridded coordinate objects.


With Random Sampling
--------------------

These functions both use standard, random `spherical point picking
<https://mathworld.wolfram.com/SpherePointPicking.html>`_ to generate angular
positions that are uniformly distributed on the surface of the unit sphere. To
retrieve angular values only, use
`~astropy.coordinates.uniform_spherical_random_surface`. For
example, to generate 4 random angular positions::

    >>> from astropy.coordinates import uniform_spherical_random_surface
    >>> pts = uniform_spherical_random_surface(size=4)
    >>> pts  # doctest: +SKIP
    <UnitSphericalRepresentation (lon, lat) in rad
        [(0.52561028, 0.38712031), (0.29900285, 0.52776066),
         (0.98199282, 0.34247723), (2.15260367, 1.01499232)]>

To generate three-dimensional positions uniformly within a spherical volume set
by a maximum radius, instead use the
`~astropy.coordinates.uniform_spherical_random_volume`
function. For example, to generate 4 random 3D positions::

    >>> from astropy.coordinates import uniform_spherical_random_volume
    >>> pts_3d = uniform_spherical_random_volume(size=4)
    >>> pts_3d  # doctest: +SKIP
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, )
        [(4.98504602, -0.74247419, 0.39752416),
         (5.53281607,  0.89425191, 0.7391255 ),
         (0.88100456,  0.21080555, 0.5531785 ),
         (6.00879324,  0.61547168, 0.61746148)]>

By default, the distance values returned are uniformly distributed within the
unit sphere (i.e., the distance values are dimensionless). To instead generate
random points within a sphere of a given dimensional radius, for example, 1
parsec, pass in a |Quantity| object with the ``max_radius`` argument::

    >>> import astropy.units as u
    >>> pts_3d = uniform_spherical_random_volume(size=4, max_radius=2*u.pc)
    >>> pts_3d  # doctest: +SKIP
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, pc)
        [(3.36590297, -0.23085809, 1.47210093),
         (6.14591179,  0.06840621, 0.9325143 ),
         (2.19194797,  0.55099774, 1.19294064),
         (5.25689272, -1.17703409, 1.63773358)]>


On a Grid
---------

No grid or lattice of points on the sphere can produce equal spacing between all
grid points, but many approximate algorithms exist for generating angular grids
with nearly even spacing (for example, `see this page
<https://bendwavy.org/pack/pack.htm>`_).

One simple and popular method in this context is the `golden spiral method
<https://stackoverflow.com/a/44164075>`_, which is available in
`astropy.coordinates` through the utility function
`~astropy.coordinates.golden_spiral_grid`. This function accepts
a single argument, ``size``, which specifies the number of points to generate in
the grid::

    >>> from astropy.coordinates import golden_spiral_grid
    >>> golden_pts = golden_spiral_grid(size=32)
    >>> golden_pts  # doctest: +FLOAT_CMP
    <UnitSphericalRepresentation (lon, lat) in rad
        [(1.94161104,  1.32014066), (5.82483312,  1.1343273 ),
         (3.42486989,  1.004232  ), (1.02490666,  0.89666582),
         (4.90812873,  0.80200278), (2.5081655 ,  0.71583806),
         (0.10820227,  0.63571129), (3.99142435,  0.56007531),
         (1.59146112,  0.48787515), (5.4746832 ,  0.41834639),
         (3.07471997,  0.35090734), (0.67475674,  0.28509644),
         (4.55797882,  0.22053326), (2.15801559,  0.15689287),
         (6.04123767,  0.09388788), (3.64127444,  0.03125509),
         (1.24131121, -0.03125509), (5.12453328, -0.09388788),
         (2.72457005, -0.15689287), (0.32460682, -0.22053326),
         (4.2078289 , -0.28509644), (1.80786567, -0.35090734),
         (5.69108775, -0.41834639), (3.29112452, -0.48787515),
         (0.89116129, -0.56007531), (4.77438337, -0.63571129),
         (2.37442014, -0.71583806), (6.25764222, -0.80200278),
         (3.85767899, -0.89666582), (1.45771576, -1.004232  ),
         (5.34093783, -1.1343273 ), (2.9409746 , -1.32014066)]>




Comparing Spherical Point Generation Methods
--------------------------------------------

.. plot::
    :align: center
    :context: close-figs

    import matplotlib.pyplot as plt
    from astropy.coordinates import uniform_spherical_random_surface, golden_spiral_grid

    fig, axes = plt.subplots(1, 2, figsize=(10, 6),
                             subplot_kw=dict(projection='3d'),
                             constrained_layout=True)

    for func, ax in zip([uniform_spherical_random_surface,
                         golden_spiral_grid], axes):
        pts = func(size=128)

        xyz = pts.to_cartesian().xyz
        ax.scatter(*xyz)

        ax.set(xlim=(-1, 1),
            ylim=(-1, 1),
            zlim=(-1, 1),
            xlabel='$x$',
            ylabel='$y$',
            zlabel='$z$')
        ax.set_title(func.__name__, fontsize=14)

    fig.suptitle('128 points', fontsize=18)