# Separations, Offsets, Catalog Matching, and Related Functionality¶

`astropy.coordinates`

contains commonly-used tools for comparing or
matching coordinate objects. Of particular importance are those for
determining separations between coordinates and those for matching a
coordinate (or coordinates) to a catalog. These are mainly implemented
as methods on the coordinate objects.

## Separations¶

The on-sky separation can be computed with the
`astropy.coordinates.BaseCoordinateFrame.separation()`

or
`astropy.coordinates.SkyCoord.separation()`

methods,
which computes the great-circle distance (*not* the small-angle approximation):

```
>>> import numpy as np
>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> c1 = SkyCoord('5h23m34.5s', '-69d45m22s', frame='icrs')
>>> c2 = SkyCoord('0h52m44.8s', '-72d49m43s', frame='fk5')
>>> sep = c1.separation(c2)
>>> sep
<Angle 20.74611448 deg>
```

The returned object is an `Angle`

instance, so it
is possible to access the angle in any of several equivalent angular
units:

```
>>> sep.radian
0.36208800460262563
>>> sep.hour
1.3830742984029318
>>> sep.arcminute
1244.7668685626384
>>> sep.arcsecond
74686.0121137583
```

Also note that the two input coordinates were not in the same frame — one is automatically converted to match the other, ensuring that even though they are in different frames, the separation is determined consistently.

In addition to the on-sky separation described above,
`astropy.coordinates.BaseCoordinateFrame.separation_3d()`

or
`astropy.coordinates.SkyCoord.separation_3d()`

methods will
determine the 3D distance between two coordinates that have `distance`

defined:

```
>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> c1 = SkyCoord('5h23m34.5s', '-69d45m22s', distance=70*u.kpc, frame='icrs')
>>> c2 = SkyCoord('0h52m44.8s', '-72d49m43s', distance=80*u.kpc, frame='icrs')
>>> sep = c1.separation_3d(c2)
>>> sep
<Distance 28.74398816 kpc>
```

## Offsets¶

Closely related to angular separations are offsets between coordinates. The key
distinction for offsets is generally the concept of a “from” and “to” coordinate
rather than the single scalar angular offset of a separation.
`coordinates`

contains conveniences to compute some of the common
offsets encountered in astronomy.

The first piece of such functionality is the
`position_angle()`

method. This method
computes the position angle between one
`SkyCoord`

instance and another (passed as the argument) following the
astronomy convention (positive angles East of North):

```
>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> c1 = SkyCoord(1*u.deg, 1*u.deg, frame='icrs')
>>> c2 = SkyCoord(2*u.deg, 2*u.deg, frame='icrs')
>>> c1.position_angle(c2).to(u.deg)
<Angle 44.97818294 deg>
```

The combination of `separation()`

and
`position_angle()`

thus give a set of
directional offsets. To do the inverse operation — determining the new
“destination” coordinate given a separation and position angle — the
`directional_offset_by()`

method is provided:

```
>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> c1 = SkyCoord(1*u.deg, 1*u.deg, frame='icrs')
>>> position_angle = 45 * u.deg
>>> separation = 1.414 * u.deg
>>> c1.directional_offset_by(position_angle, separation)
<SkyCoord (ICRS): (ra, dec) in deg
(2.0004075, 1.99964588)>
```

This technique is also useful for computing the midpoint (or indeed any point) between two coordinates in a way that accounts for spherical geometry (i.e., instead of averaging the RAs/Decs separately):

```
>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> coord1 = SkyCoord(0*u.deg, 0*u.deg, frame='icrs')
>>> coord2 = SkyCoord(1*u.deg, 1*u.deg, frame='icrs')
>>> pa = coord1.position_angle(coord2)
>>> sep = coord1.separation(coord2)
>>> coord1.directional_offset_by(pa, sep/2)
<SkyCoord (ICRS): (ra, dec) in deg
(0.49996192, 0.50001904)>
```

There is also a `spherical_offsets_to()`

method for computing angular offsets (e.g., small shifts like you might give a
telescope operator to move from a bright star to a fainter target):

```
>>> from astropy import units as u
>>> from astropy.coordinates import SkyCoord
>>> bright_star = SkyCoord('8h50m59.75s', '+11d39m22.15s', frame='icrs')
>>> faint_galaxy = SkyCoord('8h50m47.92s', '+11d39m32.74s', frame='icrs')
>>> dra, ddec = bright_star.spherical_offsets_to(faint_galaxy)
>>> dra.to(u.arcsec)
<Angle -173.78873354 arcsec>
>>> ddec.to(u.arcsec)
<Angle 10.60510342 arcsec>
```

### “Sky Offset” Frames¶

To extend the concept of spherical offsets, `coordinates`

has
a frame class `SkyOffsetFrame`

which creates distinct frames that are centered on a specific point.
These are known as “sky offset frames,” as they are a convenient way to create
a frame centered on an arbitrary position on the sky suitable for computing
positional offsets (e.g., for astrometry):

```
>>> from astropy import units as u
>>> from astropy.coordinates import SkyOffsetFrame, ICRS, SkyCoord
>>> center = ICRS(10*u.deg, 45*u.deg)
>>> center.transform_to(SkyOffsetFrame(origin=center))
<SkyOffsetICRS Coordinate (rotation=0.0 deg, origin=<ICRS Coordinate: (ra, dec) in deg
(10., 45.)>): (lon, lat) in deg
(0., 0.)>
>>> target = ICRS(11*u.deg, 46*u.deg)
>>> target.transform_to(SkyOffsetFrame(origin=center))
<SkyOffsetICRS Coordinate (rotation=0.0 deg, origin=<ICRS Coordinate: (ra, dec) in deg
(10., 45.)>): (lon, lat) in deg
(0.69474685, 1.00428706)>
```

Alternatively, the convenience method
`skyoffset_frame()`

lets you create a sky
offset frame from an existing `SkyCoord`

:

```
>>> center = SkyCoord(10*u.deg, 45*u.deg)
>>> aframe = center.skyoffset_frame()
>>> target.transform_to(aframe)
<SkyOffsetICRS Coordinate (rotation=0.0 deg, origin=<ICRS Coordinate: (ra, dec) in deg
(10., 45.)>): (lon, lat) in deg
(0.69474685, 1.00428706)>
>>> other = SkyCoord(9*u.deg, 44*u.deg, frame='fk5')
>>> other.transform_to(aframe)
<SkyCoord (SkyOffsetICRS: rotation=0.0 deg, origin=<ICRS Coordinate: (ra, dec) in deg
(10., 45.)>): (lon, lat) in deg
(-0.71943945, -0.99556216)>
```

Note

While sky offset frames *appear* to be all the same class, this not the
case: the sky offset frame for each different type of frame for `origin`

is
actually a distinct class. E.g., `SkyOffsetFrame(origin=ICRS(...))`

yields an object of class `SkyOffsetICRS`

, *not* `SkyOffsetFrame`

.
While this is not important for most uses of this class, it is important for
things like type-checking, because something like
`SkyOffsetFrame(origin=ICRS(...)).__class__ is SkyOffsetFrame`

will
*not* be `True`

, as it would be for most classes.

This same frame is also useful as a tool for defining frames that are relative to a specific, known object useful for hierarchical physical systems like galaxy groups. For example, objects around M31 are sometimes shown in a coordinate frame aligned with standard ICRA RA/Dec, but on M31:

```
>>> m31 = SkyCoord(10.6847083*u.deg, 41.26875*u.deg, frame='icrs')
>>> ngc147 = SkyCoord(8.3005*u.deg, 48.5087389*u.deg, frame='icrs')
>>> ngc147_inm31 = ngc147.transform_to(m31.skyoffset_frame())
>>> xi, eta = ngc147_inm31.lon, ngc147_inm31.lat
>>> xi
<Longitude -1.59206948 deg>
>>> eta
<Latitude 7.26183757 deg>
```

Note

Currently, distance information in the `origin`

of a
`SkyOffsetFrame`

is not
used to compute any part of the transform. The `origin`

is only used for
on-sky rotation. This may change in the future, however.

## Matching Catalogs¶

`coordinates`

leverages the coordinate framework to make it
possible to find the closest coordinates in a catalog to a desired set
of other coordinates. For example, assuming `ra1`

/`dec1`

and
`ra2`

/`dec2`

are NumPy arrays loaded from some file:

```
>>> from astropy.coordinates import SkyCoord
>>> from astropy import units as u
>>> c = SkyCoord(ra=ra1*u.degree, dec=dec1*u.degree)
>>> catalog = SkyCoord(ra=ra2*u.degree, dec=dec2*u.degree)
>>> idx, d2d, d3d = c.match_to_catalog_sky(catalog)
```

The distances returned `d3d`

are 3-dimensional distances.
Unless both source (`c`

) and catalog (`catalog`

) coordinates have
associated distances, this quantity assumes that all sources are at a distance
of 1 (dimensionless).

You can also find the nearest 3D matches, different from the on-sky
separation shown above only when the coordinates were initialized with
a `distance`

:

```
>>> c = SkyCoord(ra=ra1*u.degree, dec=dec1*u.degree, distance=distance1*u.kpc)
>>> catalog = SkyCoord(ra=ra2*u.degree, dec=dec2*u.degree, distance=distance2*u.kpc)
>>> idx, d2d, d3d = c.match_to_catalog_3d(catalog)
```

Now `idx`

are indices into `catalog`

that are the closest objects to each
of the coordinates in `c`

, `d2d`

are the on-sky distances between them, and
`d3d`

are the 3-dimensional distances. Because coordinate objects support
indexing, `idx`

enables easy access to the matched set of coordinates in
the catalog:

```
>>> matches = catalog[idx]
>>> (matches.separation_3d(c) == d3d).all()
True
>>> dra, ddec = c.spherical_offsets_to(matches)
```

This functionality can also be accessed from the
`match_coordinates_sky()`

and
`match_coordinates_3d()`

functions. These
will work on either `SkyCoord`

objects *or* the lower-level frame classes:

```
>>> from astropy.coordinates import match_coordinates_sky
>>> idx, d2d, d3d = match_coordinates_sky(c, catalog)
>>> idx, d2d, d3d = match_coordinates_sky(c.frame, catalog.frame)
```

It is possible to impose a separation constraint (e.g., the maximum separation to be
considered a match) by creating a boolean mask with `d2d`

or `d3d`

. For example:

```
>>> max_sep = 1.0 * u.arcsec
>>> idx, d2d, d3d = c.match_to_catalog_3d(catalog)
>>> sep_constraint = d2d < max_sep
>>> c_matches = c[sep_constraint]
>>> catalog_matches = catalog[idx[sep_constraint]]
```

Now, `c_matches`

and `catalog_matches`

are the matched sources in `c`

and `catalog`

, respectively, which are separated by less than 1 arcsecond.

## Searching around Coordinates¶

Closely related functionality can be used to search for *all* coordinates within
a certain distance (either 3D distance or on-sky) of another set of coordinates.
The `search_around_*`

methods (and functions) provide this functionality,
with an interface very similar to `match_coordinates_*`

:

```
>>> import numpy as np
>>> idxc, idxcatalog, d2d, d3d = catalog.search_around_sky(c, 1*u.deg)
>>> np.all(d2d < 1*u.deg)
True
```

```
>>> idxc, idxcatalog, d2d, d3d = catalog.search_around_3d(c, 1*u.kpc)
>>> np.all(d3d < 1*u.kpc)
True
```

The key difference for these methods is that there can be multiple (or no)
matches in `catalog`

around any locations in `c`

. Hence, indices into both
`c`

and `catalog`

are returned instead of just indices into `catalog`

.
These can then be indexed back into the two `SkyCoord`

objects, or, for that
matter, any array with the same order:

```
>>> np.all(c[idxc].separation(catalog[idxcatalog]) == d2d)
True
>>> np.all(c[idxc].separation_3d(catalog[idxcatalog]) == d3d)
True
>>> print(catalog_objectnames[idxcatalog])
['NGC 1234' 'NGC 4567' ...]
```

Note, though, that this dual-indexing means that `search_around_*`

does not
work well if one of the coordinates is a scalar, because the returned index
would not make sense for a scalar:

```
>>> scalarc = SkyCoord(ra=1*u.deg, dec=2*u.deg, distance=distance1*u.kpc)
>>> idxscalarc, idxcatalog, d2d, d3d = catalog.search_around_sky(scalarc, 1*u.deg)
ValueError: One of the inputs to search_around_sky is a scalar.
```

As a result (and because the `search_around_*`

algorithm is inefficient in
the scalar case), the best approach for this scenario is to instead
use the `separation*`

methods:

```
>>> d2d = scalarc.separation(catalog)
>>> catalogmsk = d2d < 1*u.deg
>>> d3d = scalarc.separation_3d(catalog)
>>> catalog3dmsk = d3d < 1*u.kpc
```

The resulting `catalogmsk`

or `catalog3dmsk`

variables are boolean arrays
rather than arrays of indices, but in practice they usually can be used in
the same way as `idxcatalog`

from the above examples. If you definitely do
need indices instead of boolean masks, you can do:

```
>>> idxcatalog = np.where(catalogmsk)[0]
>>> idxcatalog3d = np.where(catalog3dmsk)[0]
```