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### B.1.19 Tilings

Pixel tiling functions for the celestial sphere.

The `k` parameter for the HEALPix functions is the HEALPix order, which can be in the range 0<=k<=29. This is the logarithm to base 2 of the HEALPix NSIDE parameter. At order `k`, there are 12*4^k pixels on the sphere.

`htmIndex( level, lon, lat )`
Gives the HTM (Hierachical Triangular Mesh) pixel index for a given sky position.

• Parameters:
• `level` (integer): HTM level
• `lon` (floating point): longitude in degrees
• `lat` (floating point): latitude in degrees
• Return value
• (long integer): pixel index

`healpixNestIndex( k, lon, lat )`
Gives the pixel index for a given sky position in the HEALPix NEST scheme.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `lon` (floating point): longitude in degrees
• `lat` (floating point): latitude in degrees
• Return value
• (long integer): pixel index

`healpixRingIndex( k, lon, lat )`
Gives the pixel index for a given sky position in the HEALPix RING scheme.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `lon` (floating point): longitude in degrees
• `lat` (floating point): latitude in degrees
• Return value
• (long integer): pixel index

`healpixNestLon( k, index )`
Returns the longitude of the approximate center of the tile with a given index in the HEALPix NEST scheme.

Note: the `index` parameter is 0-based, unlike the table row index special token `\$index` (a.k.a. `\$0`), which is 1-based. So if the HEALpix index is implicitly determined by the table row, the value `\$index-1` should be used.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `index` (long integer): healpix index
• Return value
• (floating point): longitude in degrees

`healpixNestLat( k, index )`
Returns the latitude of the approximate center of the tile with a given index in the HEALPix NEST scheme.

Note: the `index` parameter is 0-based, unlike the table row index special token `\$index` (a.k.a. `\$0`), which is 1-based. So if the HEALpix index is implicitly determined by the table row, the value `\$index-1` should be used.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `index` (long integer): healpix index
• Return value
• (floating point): latitude in degrees

`healpixRingLon( k, index )`
Returns the longitude of the approximate center of the tile with a given index in the HEALPix RING scheme.

Note: the `index` parameter is 0-based, unlike the table row index special token `\$index` (a.k.a. `\$0`), which is 1-based. So if the HEALpix index is implicitly determined by the table row, the value `\$index-1` should be used.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `index` (long integer): healpix index
• Return value
• (floating point): longitude in degrees

`healpixRingLat( k, index )`
Returns the latitude of the approximate center of the tile with a given index in the HEALPix RING scheme.

Note: the `index` parameter is 0-based, unlike the table row index special token `\$index` (a.k.a. `\$0`), which is 1-based. So if the HEALpix index is implicitly determined by the table row, the value `\$index-1` should be used.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `index` (long integer): healpix index
• Return value
• (floating point): latitude in degrees

`healpixNestToRing( k, nestIndex )`
Converts a healpix tile index from the NEST to the RING scheme at a given order.

Note: the `nestIndex` parameter is 0-based, unlike the table row index special token `\$index` (a.k.a. `\$0`), which is 1-based. So if the HEALpix index is implicitly determined by the table row, the value `\$index-1` should be used.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `nestIndex` (long integer): pixel index in NEST scheme
• Return value
• (long integer): pixel index in RING scheme

`healpixRingToNest( k, ringIndex )`
Converts a healpix tile index from the RING to the NEST scheme at a given order.

Note: the `ringIndex` parameter is 0-based, unlike the table row index special token `\$index` (a.k.a. `\$0`), which is 1-based. So if the HEALpix index is implicitly determined by the table row, the value `\$index-1` should be used.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• `ringIndex` (long integer): pixel index in RING scheme
• Return value
• (long integer): pixel index in NEST scheme

`healpixK( pixelsize )`
Gives the HEALPix resolution parameter suitable for a given pixel size. This `k` value (also variously known as order, level, depth) is the logarithm to base 2 of the Nside parameter.

• Parameters:
• `pixelsize` (floating point): pixel size in degrees
• Return value
• (integer): HEALPix order `k`

`healpixResolution( k )`
Gives the approximate resolution in degrees for a given HEALPix resolution parameter `k` This `k` value is the logarithm to base 2 of the Nside parameter.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• Return value
• (floating point): approximate angular resolution in degrees

`healpixSteradians( k )`
Returns the solid angle in steradians of each HEALPix pixel at a given order.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• Return value
• (floating point): pixel size in steradians
• Examples:
• `healpixSteradians(5) = 0.0010226538585904274`
• `4*PI/healpixSteradians(0) = 12.0`

`healpixSqdeg( k )`
Returns the solid angle in square degrees of each HEALPix pixel at a given order.

• Parameters:
• `k` (integer): HEALPix order (0..29)
• Return value
• (floating point): pixel size in steradians
• Examples:
• `healpixSqdeg(5) = 3.357174580844667`
• `round(12 * healpixSqdeg(0)) = 41253`

`steradiansToSqdeg( sr )`
Converts a solid angle from steradians to square degrees.

The unit sphere is 4*PI steradians = 360*360/PI square degrees.

• Parameters:
• `sr` (floating point): quantity in steradians
• Return value
• (floating point): quantity in sqare degrees
• Example:
• `round(steradiansToSqdeg(4*PI)) = 41253`

`sqdegToSteradians( sqdeg )`
Converts a solid angle from square degrees to steradians.

The unit sphere is 4*PI steradians = 360*360/PI square degrees.

• Parameters:
• `sqdeg` (floating point): quantity in square degrees
• Return value
• (floating point): quantity in steradians
• Example:
• `round(sqdegToSteradians(41253)/PI) = 4`

`htmLevel( pixelsize )`
Gives the HTM `level` parameter suitable for a given pixel size.

• Parameters:
• `pixelsize` (floating point): required resolution in degrees
• Return value
• (integer): HTM level parameter

`htmResolution( level )`
Gives the approximate resolution in degrees for a given HTM depth level.

• Parameters:
• `level` (integer): HTM depth
• Return value
• (floating point): approximate angular resolution in degrees

`SQDEG`
Solid angle in steradians corresponding to 1 square degree.

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