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- Using and Designing Coordinate Representations

# 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:

- `CartesianRepresentation`: Cartesian coordinates `x`, `y`, and `z`.

- `SphericalRepresentation`: spherical polar coordinates represented by a longitude (`lon`), a latitude (`lat`), and a distance (`distance`). The latitude is a value ranging from -90 to 90 degrees.

- `UnitSphericalRepresentation`: spherical polar coordinates on a unit sphere, represented by a longitude (`lon`) and latitude (`lat`).

- `PhysicsSphericalRepresentation`: spherical polar coordinates, represented by an inclination (`theta`) and azimuthal angle (`phi`), and radius `r`. The inclination goes from 0 to 180 degrees, and is related to the latitude in the `SphericalRepresentation` by `theta = 90 deg - lat`.

- `CylindricalRepresentation`: cylindrical polar coordinates, represented by a cylindrical radius (`rho`), azimuthal angle (`phi`), and height (`z`).

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
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (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
    <UnitSphericalRepresentation (lon, lat) in rad
        (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`. Corresponding functions, as well as others that affect the shape, such as `atleast_1d` and `rollaxis`, work as expected.

### Example

To pass array `Quantity` objects to representations:

    >>> 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.)]>

To manipulate using methods and `numpy` functions:

    >>> 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.)]]>
    >>> car_array[2]
    <CartesianRepresentation (x, y, z) in m
        (2., 12., 22.)>
    >>> car_array[2] = car_array[1]
    >>> car_array[:3]
    <CartesianRepresentation (x, y, z) in m
        [(0., 10., 20.), (1., 11., 21.), (1., 11., 21.)]>
    >>> np.roll(car_array, 1)
    <CartesianRepresentation (x, y, z) in m
        [(5., 15., 25.), (0., 10., 20.), (1., 11., 21.), (1., 11., 21.),
         (3., 13., 23.), (4., 14., 24.)]>

And to set elements using other representation classes (as long as they are compatible in their units and number of dimensions):

    >>> car_array[2] = SphericalRepresentation(0*u.deg, 0*u.deg, 99*u.m)
    >>> car_array[:3]
    <CartesianRepresentation (x, y, z) in m
        [(0., 10., 20.), (1., 11., 21.), (99., 0., 0.)]>
    >>> car_array[0] = UnitSphericalRepresentation(0*u.deg, 0*u.deg)
    Traceback (most recent call last):
    ...
    ValueError: value must be representable as CartesianRepresentation without loss of information.

## 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.

### Examples

To see how vector arithmetic operations work with representation objects, 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.)

### Examples

To see how the `Differential` classes of representations 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
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        [(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
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (1.57079633, 0., 2.)>
    >>> sph_coo + (big_offset + big_offset)
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (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
    >>> radvel = RadialDifferential(1000*u.km/u.s)
    >>> sph_coo + radvel * 1. * u.Myr
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (0., 0., 2.02271217)>
    >>> pm = UnitSphericalDifferential(1.*u.mas/u.yr, 0.*u.mas/u.yr)
    >>> sph_coo + pm * 1. * u.Myr
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (0.0048481, 0., 1.00001175)>
    >>> pm + radvel
    <SphericalDifferential (d_lon, d_lat, d_distance) in (mas / yr, mas / yr, km / s)
        (1., 0., 1000.)>
    >>> sph_coo + (pm + radvel) * 1. * u.Myr
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (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
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (0.0048481, 1.04719246, 1.00000294)>
    >>> sph_lat60 + pm_coslat * 1. * u.Myr
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (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
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, kpc)
        (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

`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.

### 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).

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)
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, AU)
        [(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
    <SphericalRepresentation (lon, lat, distance) in (rad, rad, AU)
      [(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.)]>

## Creating Your Own Representations

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 (`dict`):

  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 = {
            "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