# Coordinate Transforms¶

## Numpy¶

 radec_to_lm(radec[, phase_centre]) Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre. radec_to_lmn(radec[, phase_centre]) Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre. lm_to_radec(lm[, phase_centre]) Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre. lmn_to_radec(lmn[, phase_centre]) Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre.

Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre.

\begin{eqnarray} & l =& \, \cos \, \delta \sin \, \Delta \alpha \\ & m =& \, \sin \, \delta \cos \, \delta 0 - \cos \delta \sin \delta 0 \cos \Delta \alpha \\ & n =& \, \sqrt{1 - l^2 - m^2} - 1 \end{eqnarray}

where $$\Delta \alpha = \alpha - \alpha 0$$ is the difference between the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
radecnumpy.ndarray

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

phase_centrenumpy.ndarray, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
numpy.ndarray

lm Direction Cosines of shape (coord, 2)

Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre.

\begin{eqnarray} & l =& \, \cos \, \delta \sin \, \Delta \alpha \\ & m =& \, \sin \, \delta \cos \, \delta 0 - \cos \delta \sin \delta 0 \cos \Delta \alpha \\ & n =& \, \sqrt{1 - l^2 - m^2} - 1 \end{eqnarray}

where $$\Delta \alpha = \alpha - \alpha 0$$ is the difference between the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
radecnumpy.ndarray

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

phase_centrenumpy.ndarray, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
numpy.ndarray

lm Direction Cosines of shape (coord, 3)

Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre.

\begin{eqnarray} & \delta = & \, \arcsin \left( m \cos \delta 0 + n \sin \delta 0 \right) \\ & \alpha = & \, \arctan \left( \frac{l}{n \cos \delta 0 - m \sin \delta 0} \right) \\ \end{eqnarray}

where $$\alpha$$ is the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
lmnumpy.ndarray

lm Direction Cosines of shape (coord, 2)

phase_centrenumpy.ndarray, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
numpy.ndarray

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre.

\begin{eqnarray} & \delta = & \, \arcsin \left( m \cos \delta 0 + n \sin \delta 0 \right) \\ & \alpha = & \, \arctan \left( \frac{l}{n \cos \delta 0 - m \sin \delta 0} \right) \\ \end{eqnarray}

where $$\alpha$$ is the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
lmnnumpy.ndarray

lm Direction Cosines of shape (coord, 3)

phase_centrenumpy.ndarray, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
numpy.ndarray

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

 radec_to_lm(radec[, phase_centre]) Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre. radec_to_lmn(radec[, phase_centre]) Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre. lm_to_radec(lm[, phase_centre]) Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre. lmn_to_radec(lmn[, phase_centre]) Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre.

Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre.

\begin{eqnarray} & l =& \, \cos \, \delta \sin \, \Delta \alpha \\ & m =& \, \sin \, \delta \cos \, \delta 0 - \cos \delta \sin \delta 0 \cos \Delta \alpha \\ & n =& \, \sqrt{1 - l^2 - m^2} - 1 \end{eqnarray}

where $$\Delta \alpha = \alpha - \alpha 0$$ is the difference between the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
radecdask.array.Array

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

phase_centredask.array.Array, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
dask.array.Array

lm Direction Cosines of shape (coord, 2)

Converts Right-Ascension/Declination coordinates in radians to a Direction Cosine lm coordinates, relative to the Phase Centre.

\begin{eqnarray} & l =& \, \cos \, \delta \sin \, \Delta \alpha \\ & m =& \, \sin \, \delta \cos \, \delta 0 - \cos \delta \sin \delta 0 \cos \Delta \alpha \\ & n =& \, \sqrt{1 - l^2 - m^2} - 1 \end{eqnarray}

where $$\Delta \alpha = \alpha - \alpha 0$$ is the difference between the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
radecdask.array.Array

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

phase_centredask.array.Array, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
dask.array.Array

lm Direction Cosines of shape (coord, 3)

Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre.

\begin{eqnarray} & \delta = & \, \arcsin \left( m \cos \delta 0 + n \sin \delta 0 \right) \\ & \alpha = & \, \arctan \left( \frac{l}{n \cos \delta 0 - m \sin \delta 0} \right) \\ \end{eqnarray}

where $$\alpha$$ is the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
lmdask.array.Array

lm Direction Cosines of shape (coord, 2)

phase_centredask.array.Array, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
dask.array.Array

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.

Convert Direction Cosine lm coordinates to Right Ascension/Declination coordinates in radians, relative to the Phase Centre.

\begin{eqnarray} & \delta = & \, \arcsin \left( m \cos \delta 0 + n \sin \delta 0 \right) \\ & \alpha = & \, \arctan \left( \frac{l}{n \cos \delta 0 - m \sin \delta 0} \right) \\ \end{eqnarray}

where $$\alpha$$ is the Right Ascension of each coordinate and the phase centre and $$\delta 0$$ is the Declination of the phase centre.

Parameters
lmndask.array.Array

lm Direction Cosines of shape (coord, 3)

phase_centredask.array.Array, optional

radec coordinates of the Phase Centre. Shape (2,)

Returns
dask.array.Array

radec coordinates of shape (coord, 2) where Right-Ascension and Declination are in the last 2 components, respectively.