# Using and Designing Coordinate Representations¶

Points in a 3D vector space can be represented in different ways, such as Cartesian, spherical polar, cylindrical, and so on. These underlie the way coordinate data in astropy.coordinates is represented, as described in the Overview of astropy.coordinates Concepts. Below, we describe how you can use them on their own as a way to convert between different representations, including ones not built-in, and to do simple vector arithmetic.

The built-in representation classes are:

Note

For information about using and changing the representation of SkyCoord objects, see the Representations section.

## Instantiating and Converting¶

Representation classes are instantiated with Quantity objects:

>>> from astropy import units as u
>>> from astropy.coordinates.representation import CartesianRepresentation
>>> car = CartesianRepresentation(3 * u.kpc, 5 * u.kpc, 4 * u.kpc)
>>> car
<CartesianRepresentation (x, y, z) in kpc
(3., 5., 4.)>


Array Quantity objects can also be passed to representations. They will have the expected shape, which can be changed using methods with the same names as those for ndarray, such as reshape, ravel, etc.:

>>> x = u.Quantity([[1., 0., 0.], [3., 5., 3.]], u.m)
>>> y = u.Quantity([[0., 2., 0.], [4., 0., -4.]], u.m)
>>> z = u.Quantity([[0., 0., 3.], [0., 12., -12.]], u.m)
>>> car_array = CartesianRepresentation(x, y, z)
>>> car_array
<CartesianRepresentation (x, y, z) in m
[[(1.,  0.,   0.), (0.,  2.,   0.), (0.,  0.,   3.)],
[(3.,  4.,   0.), (5.,  0.,  12.), (3., -4., -12.)]]>
>>> car_array.shape
(2, 3)
>>> car_array.ravel()
<CartesianRepresentation (x, y, z) in m
[(1.,  0.,   0.), (0.,  2.,   0.), (0.,  0.,   3.), (3.,  4.,   0.),
(5.,  0.,  12.), (3., -4., -12.)]>


Representations can be converted to other representations using the represent_as method:

>>> from astropy.coordinates.representation import SphericalRepresentation, CylindricalRepresentation
>>> sph = car.represent_as(SphericalRepresentation)
>>> sph
(1.03037683, 0.60126422, 7.07106781)>
>>> cyl = car.represent_as(CylindricalRepresentation)
>>> cyl
<CylindricalRepresentation (rho, phi, z) in (kpc, rad, kpc)
(5.83095189, 1.03037683, 4.)>


All representations can be converted to each other without loss of information, with the exception of UnitSphericalRepresentation. This class is used to store the longitude and latitude of points but does not contain any distance to the points, and assumes that they are located on a unit and dimensionless sphere:

>>> from astropy.coordinates.representation import UnitSphericalRepresentation
>>> sph_unit = car.represent_as(UnitSphericalRepresentation)
>>> sph_unit
(1.03037683, 0.60126422)>


Converting back to Cartesian, the absolute scaling information has been removed, and the points are still located on a unit sphere:

>>> sph_unit = car.represent_as(UnitSphericalRepresentation)
>>> sph_unit.represent_as(CartesianRepresentation)
<CartesianRepresentation (x, y, z) [dimensionless]
(0.42426407, 0.70710678, 0.56568542)>


## Array Values and NumPy Array Method Analogs¶

Array Quantity objects can also be passed to representations, and such representations can be sliced, reshaped, etc., using the same methods as are available to ndarray:

>>> import numpy as np
>>> x = np.linspace(0., 5., 6)
>>> y = np.linspace(10., 15., 6)
>>> z = np.linspace(20., 25., 6)
>>> car_array = CartesianRepresentation(x * u.m, y * u.m, z * u.m)
>>> car_array
<CartesianRepresentation (x, y, z) in m
[(0., 10., 20.), (1., 11., 21.), (2., 12., 22.),
(3., 13., 23.), (4., 14., 24.), (5., 15., 25.)]>
>>> car_array[2]
<CartesianRepresentation (x, y, z) in m
(2., 12., 22.)>
>>> car_array.reshape(3, 2)
<CartesianRepresentation (x, y, z) in m
[[(0., 10., 20.), (1., 11., 21.)],
[(2., 12., 22.), (3., 13., 23.)],
[(4., 14., 24.), (5., 15., 25.)]]>


## Vector Arithmetic¶

Representations support basic vector arithmetic such as taking the norm, multiplying with and dividing by quantities, and taking dot and cross products, as well as adding, subtracting, summing and taking averages of representations, and multiplying with matrices.

Note

All arithmetic except the matrix multiplication works with non-Cartesian representations as well. For taking the norm, multiplication, and division, this uses just the non-angular components, while for the other operations the representation is converted to Cartesian internally before the operation is done, and the result is converted back to the original representation. Hence, for optimal speed it may be best to work using Cartesian representations.

To see how the operations work, consider the following examples:

>>> car_array = CartesianRepresentation([[1., 0., 0.], [3., 5.,  3.]] * u.m,
...                                     [[0., 2., 0.], [4., 0., -4.]] * u.m,
...                                     [[0., 0., 3.], [0.,12.,-12.]] * u.m)
>>> car_array
<CartesianRepresentation (x, y, z) in m
[[(1.,  0.,  0.), (0.,  2.,   0.), (0.,  0.,   3.)],
[(3.,  4.,  0.), (5.,  0.,  12.), (3., -4., -12.)]]>
>>> car_array.norm()
<Quantity [[ 1.,  2.,  3.],
[ 5., 13., 13.]] m>
>>> car_array / car_array.norm()
<CartesianRepresentation (x, y, z) [dimensionless]
[[(1.        ,  0.        ,  0.        ),
(0.        ,  1.        ,  0.        ),
(0.        ,  0.        ,  1.        )],
[(0.6       ,  0.8       ,  0.        ),
(0.38461538,  0.        ,  0.92307692),
(0.23076923, -0.30769231, -0.92307692)]]>
>>> (car_array[1] - car_array[0]) / (10. * u.s)
<CartesianRepresentation (x, y, z) in m / s
[(0.2,  0.4,  0. ), (0.5, -0.2,  1.2), (0.3, -0.4, -1.5)]>
>>> car_array.sum()
<CartesianRepresentation (x, y, z) in m
(12.,  2.,  3.)>
>>> car_array.mean(axis=0)
<CartesianRepresentation (x, y, z) in m
[(2. ,  2.,  0. ), (2.5,  1.,  6. ), (1.5, -2., -4.5)]>

>>> unit_x = UnitSphericalRepresentation(0.*u.deg, 0.*u.deg)
>>> unit_y = UnitSphericalRepresentation(90.*u.deg, 0.*u.deg)
>>> unit_z = UnitSphericalRepresentation(0.*u.deg, 90.*u.deg)
>>> car_array.dot(unit_x)
<Quantity [[1., 0., 0.],
[3., 5., 3.]] m>
>>> car_array.dot(unit_y)
<Quantity [[ 6.12323400e-17,  2.00000000e+00,  0.00000000e+00],
[ 4.00000000e+00,  3.06161700e-16, -4.00000000e+00]] m>
>>> car_array.dot(unit_z)
<Quantity [[ 6.12323400e-17,  0.00000000e+00,  3.00000000e+00],
[ 1.83697020e-16,  1.20000000e+01, -1.20000000e+01]] m>
>>> car_array.cross(unit_x)
<CartesianRepresentation (x, y, z) in m
[[(0.,  0.,  0.), (0.,   0., -2.), (0.,   3.,  0.)],
[(0.,  0., -4.), (0.,  12.,  0.), (0., -12.,  4.)]]>

>>> from astropy.coordinates.matrix_utilities import rotation_matrix
>>> rotation = rotation_matrix(90 * u.deg, axis='z')
>>> rotation
array([[ 6.12323400e-17,  1.00000000e+00,  0.00000000e+00],
[-1.00000000e+00,  6.12323400e-17,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
>>> car_array.transform(rotation)
<CartesianRepresentation (x, y, z) in m
[[( 6.12323400e-17, -1.00000000e+00,   0.),
( 2.00000000e+00,  1.22464680e-16,   0.),
( 0.00000000e+00,  0.00000000e+00,   3.)],
[( 4.00000000e+00, -3.00000000e+00,   0.),
( 3.06161700e-16, -5.00000000e+00,  12.),
(-4.00000000e+00, -3.00000000e+00, -12.)]]>


## Differentials and Derivatives of Representations¶

In addition to positions in 3D space, coordinates also deal with proper motions and radial velocities, which require a way to represent differentials of coordinates (i.e., finite realizations) of derivatives. To support this, the representations all have corresponding Differential classes, which can hold offsets or derivatives in terms of the components of the representation class. Adding such an offset to a representation means the offset is taken in the direction of the corresponding coordinate. (Although for any representation other than Cartesian, this is only defined relative to a specific location, as the unit vectors are not invariant.)

To see how this works, consider the following:

>>> from astropy.coordinates import SphericalRepresentation, SphericalDifferential
>>> sph_coo = SphericalRepresentation(lon=0.*u.deg, lat=0.*u.deg,
...                                   distance=1.*u.kpc)
>>> sph_derivative = SphericalDifferential(d_lon=1.*u.arcsec/u.yr,
...                                        d_lat=0.*u.arcsec/u.yr,
...                                        d_distance=0.*u.km/u.s)
>>> sph_derivative.to_cartesian(base=sph_coo)
<CartesianRepresentation (x, y, z) in arcsec kpc / (rad yr)
(0., 1., 0.)>


Note how the conversion to Cartesian can only be done using a base, since otherwise the code cannot know what direction an increase in longitude corresponds to. For lon=0, this is in the y direction. Now, to get the coordinates at two later times:

>>> sph_coo + sph_derivative * [1., 3600*180/np.pi] * u.yr
[(4.84813681e-06, 0., 1.        ), (7.85398163e-01, 0., 1.41421356)]>


The above shows how addition is not to longitude itself, but in the direction of increasing longitude: for the large shift, by the equivalent of one radian, the distance has increased as well (after all, a source will likely not move along a curve on the sky!). This also means that the order of operations is important:

>>> big_offset = SphericalDifferential(1.*u.radian, 0.*u.radian, 0.*u.kpc)
>>> sph_coo + big_offset + big_offset
(1.57079633, 0., 2.)>
>>> sph_coo + (big_offset + big_offset)
(1.10714872, 0., 2.23606798)>


Often, you may have just a proper motion or a radial velocity, but not both:

>>> from astropy.coordinates import UnitSphericalDifferential, RadialDifferential
>>> sph_coo + radvel * 1. * u.Myr
(0., 0., 2.02271217)>
>>> pm = UnitSphericalDifferential(1.*u.mas/u.yr, 0.*u.mas/u.yr)
>>> sph_coo + pm * 1. * u.Myr
(0.0048481, 0., 1.00001175)>
<SphericalDifferential (d_lon, d_lat, d_distance) in (mas / yr, mas / yr, km / s)
(1., 0., 1000.)>
>>> sph_coo + (pm + radvel) * 1. * u.Myr
(0.00239684, 0., 2.02271798)>


Note in the above that the proper motion is defined strictly as a change in longitude (i.e., it does not include a cos(latitude) term). There are special classes where this term is included:

>>> from astropy.coordinates import UnitSphericalCosLatDifferential
>>> sph_lat60 = SphericalRepresentation(lon=0.*u.deg, lat=60.*u.deg,
...                                     distance=1.*u.kpc)
>>> pm = UnitSphericalDifferential(1.*u.mas/u.yr, 0.*u.mas/u.yr)
>>> pm
<UnitSphericalDifferential (d_lon, d_lat) in mas / yr
(1., 0.)>
>>> pm_coslat = UnitSphericalCosLatDifferential(1.*u.mas/u.yr, 0.*u.mas/u.yr)
>>> pm_coslat
<UnitSphericalCosLatDifferential (d_lon_coslat, d_lat) in mas / yr
(1., 0.)>
>>> sph_lat60 + pm * 1. * u.Myr
(0.0048481, 1.04719246, 1.00000294)>
>>> sph_lat60 + pm_coslat * 1. * u.Myr
(0.00969597, 1.0471772, 1.00001175)>


Close inspections shows that indeed the changes are as expected. The systems with and without cos(latitude) can be converted to each other, provided you supply the base (representation):

>>> usph_lat60 = sph_lat60.represent_as(UnitSphericalRepresentation)
>>> pm_coslat2 = pm.represent_as(UnitSphericalCosLatDifferential,
...                              base=usph_lat60)
>>> pm_coslat2
<UnitSphericalCosLatDifferential (d_lon_coslat, d_lat) in mas / yr
(0.5, 0.)>
>>> sph_lat60 + pm_coslat2 * 1. * u.Myr
(0.0048481, 1.04719246, 1.00000294)>


Note

At present, the differential classes are generally meant to work with first derivatives, but they do not check the units of the inputs to enforce this. Passing in second derivatives (e.g., acceleration values with acceleration units) will succeed, but any transformations that occur through re-representation of the differential will not necessarily be correct.

## Attaching Differential Objects to Representation Objects¶

Warning

The API for this functionality may change in future versions and should be viewed as provisional!

Differential objects can be attached to Representation objects as a way to encapsulate related information into a single object. Differential objects can be passed in to the initializer of any of the built-in Representation classes. For example, to store a single velocity differential with a position:

>>> from astropy.coordinates import representation as r
>>> dif = r.SphericalDifferential(d_lon=1 * u.mas/u.yr,
...                               d_lat=2 * u.mas/u.yr,
...                               d_distance=3 * u.km/u.s)
>>> rep = r.SphericalRepresentation(lon=0.*u.deg, lat=0.*u.deg,
...                                 distance=1.*u.kpc,
...                                 differentials=dif)
>>> rep
<SphericalRepresentation (lon, lat, distance) in (deg, deg, kpc)
(0., 0., 1.)
(has differentials w.r.t.: 's')>
>>> rep.differentials
{'s': <SphericalDifferential (d_lon, d_lat, d_distance) in (mas / yr, mas / yr, km / s)
(1., 2., 3.)>}


The Differential objects are stored as a Python dictionary on the Representation object with keys equal to the (string) unit with which the differential derivatives are taken (converted to SI). For example, in this case the key is 's' (second) because the Differential units are velocities, a time derivative. Passing a single differential to the Representation initializer will automatically generate the necessary key and store it in the differentials dictionary, but a dictionary is required to specify multiple differentials:

>>> dif2 = r.SphericalDifferential(d_lon=4 * u.mas/u.yr**2,
...                                d_lat=5 * u.mas/u.yr**2,
...                                d_distance=6 * u.km/u.s**2)
>>> rep = r.SphericalRepresentation(lon=0.*u.deg, lat=0.*u.deg,
...                                 distance=1.*u.kpc,
...                                 differentials={'s': dif, 's2': dif2})
>>> rep.differentials['s']
<SphericalDifferential (d_lon, d_lat, d_distance) in (mas / yr, mas / yr, km / s)
(1., 2., 3.)>
>>> rep.differentials['s2']
<SphericalDifferential (d_lon, d_lat, d_distance) in (mas / yr2, mas / yr2, km / s2)
(4., 5., 6.)>


Differential objects can also be attached to a Representation after creation:

>>> rep = r.CartesianRepresentation(x=1 * u.kpc, y=2 * u.kpc, z=3 * u.kpc)
>>> dif = r.CartesianDifferential(*[1, 2, 3] * u.km/u.s)
>>> rep = rep.with_differentials(dif)
>>> rep
<CartesianRepresentation (x, y, z) in kpc
(1., 2., 3.)
(has differentials w.r.t.: 's')>


This works for array data as well, as long as the shape of the Differential data is the same as that of the Representation:

>>> xyz = np.arange(12).reshape(3, 4) * u.au
>>> d_xyz = np.arange(12).reshape(3, 4) * u.km/u.s
>>> rep = r.CartesianRepresentation(*xyz)
>>> dif = r.CartesianDifferential(*d_xyz)
>>> rep = rep.with_differentials(dif)
>>> rep
<CartesianRepresentation (x, y, z) in AU
[(0., 4.,  8.), (1., 5.,  9.), (2., 6., 10.), (3., 7., 11.)]
(has differentials w.r.t.: 's')>


As with a Representation instance without a differential, to convert the positional data to a new representation, use the .represent_as():

>>> rep.represent_as(r.SphericalRepresentation)
[(1.57079633, 1.10714872,  8.94427191),
(1.37340077, 1.05532979, 10.34408043),
(1.24904577, 1.00685369, 11.83215957),
(1.16590454, 0.96522779, 13.37908816)]>


However, by passing just the desired representation class, only the Representation has changed, and the differentials are dropped. To re-represent both the Representation and any Differential objects, you must specify target classes for the Differential as well:

>>> rep2 = rep.represent_as(r.SphericalRepresentation, r.SphericalDifferential)
>>> rep2
[(1.57079633, 1.10714872,  8.94427191),
(1.37340077, 1.05532979, 10.34408043),
(1.24904577, 1.00685369, 11.83215957),
(1.16590454, 0.96522779, 13.37908816)]
(has differentials w.r.t.: 's')>
>>> rep2.differentials['s']
<SphericalDifferential (d_lon, d_lat, d_distance) in (km rad / (AU s), km rad / (AU s), km / s)
[( 6.12323400e-17, 1.11022302e-16,  8.94427191),
(-2.77555756e-17, 5.55111512e-17, 10.34408043),
( 0.00000000e+00, 0.00000000e+00, 11.83215957),
( 5.55111512e-17, 0.00000000e+00, 13.37908816)]>


Shape-changing operations (e.g., reshapes) are propagated to all Differential objects because they are guaranteed to have the same shape as their host Representation object:

>>> rep.shape
(4,)
>>> rep.differentials['s'].shape
(4,)
>>> new_rep = rep.reshape(2, 2)
>>> new_rep.shape
(2, 2)
>>> new_rep.differentials['s'].shape
(2, 2)


This also works for slicing:

>>> new_rep = rep[:2]
>>> new_rep.shape
(2,)
>>> new_rep.differentials['s'].shape
(2,)


Operations on representations that return Quantity objects (as opposed to other Representation instances) still work, but only operate on the positional information, for example:

>>> rep.norm()
<Quantity [ 8.94427191, 10.34408043, 11.83215957, 13.37908816] AU>


Operations that involve combining or scaling representations or pairs of representation objects that contain differentials will currently fail, but support for some operations may be added in future versions:

>>> rep + rep
Traceback (most recent call last):
...
TypeError: Operation 'add' is not supported when differentials are attached to a CartesianRepresentation.


If you have a Representation with attached Differential objects, you can retrieve a copy of the Representation without the Differential object and use this Differential-free object for any arithmetic operation:

>>> 15 * rep.without_differentials()
<CartesianRepresentation (x, y, z) in AU
[( 0.,  60., 120.), (15.,  75., 135.), (30.,  90., 150.),
(45., 105., 165.)]>


To create your own representation class, your class must inherit from the BaseRepresentation class. This base has an __init__ method that will put all arguments components through their initializers, verify they can be broadcast against each other, and store the components on self as the name prefixed with ‘_’. Furthermore, through its metaclass it provides default properties for the components so that they can be accessed using <instance>.<component>. For the machinery to work, the following must be defined:

• attr_classes class attribute (OrderedDict):

Defines through its keys the names of the components (as well as the default order), and through its values defines the class of which they should be instances (which should be Quantity or a subclass, or anything that can initialize it).

• from_cartesian class method:

Takes a CartesianRepresentation object and returns an instance of your class.

• to_cartesian method:

Returns a CartesianRepresentation object.

• __init__ method (optional):

If you want more than the basic initialization and checks provided by the base representation class, or just an explicit signature, you can define your own __init__. In general, it is recommended to stay close to the signature assumed by the base representation, __init__(self, comp1, comp2, comp3, copy=True), and use super to call the base representation initializer.

Once you do this, you will then automatically be able to call represent_as to convert other representations to/from your representation class. Your representation will also be available for use in SkyCoord and all frame classes.

A representation class may also have a _unit_representation attribute (although it is not required). This attribute points to the appropriate “unit” representation (i.e., a representation that is dimensionless). This is probably only meaningful for subclasses of SphericalRepresentation, where it is assumed that it will be a subclass of UnitSphericalRepresentation.

Finally, if you wish to also use offsets in your coordinate system, two further methods should be defined (please see SphericalRepresentation for an example):

• unit_vectors method:

Returns a dict with a CartesianRepresentation of unit vectors in the direction of each component.

• scale_factors method:

Returns a dict with a Quantity for each component with the appropriate physical scale factor for a unit change in that direction.

And furthermore you should define a Differential class based on BaseDifferential. This class only needs to define:

• base_representation attribute:

A link back to the representation for which this differential holds.

In pseudo-code, this means that a class will look like:

class MyRepresentation(BaseRepresentation):

attr_classes = OrderedDict([('comp1', ComponentClass1),
('comp2', ComponentClass2),
('comp3', ComponentClass3)])

# __init__ is optional
def __init__(self, comp1, comp2, comp3, copy=True):
super().__init__(comp1, comp2, comp3, copy=copy)
...

@classmethod
def from_cartesian(self, cartesian):
...
return MyRepresentation(...)

def to_cartesian(self):
...
return CartesianRepresentation(...)

# if differential motion is needed
def unit_vectors(self):
...
return {'comp1': CartesianRepresentation(...),
'comp2': CartesianRepresentation(...),
'comp3': CartesianRepresentation(...)}

def scale_factors(self):
...
return {'comp1': ...,
'comp2': ...,
'comp3': ...}

class MyDifferential(BaseDifferential):
base_representation = MyRepresentation