Coordinate systems

Every FITS image involves two coordinate systems:

Pixel coordinates or image coordinates index pixels in the image as an array with integral indices. Pixel coordinates are given as a tuple (X, Y). Pixel coordinates typically start at (0, 0), which is typically plotted at the lower-left corner of the image, with the X coordinate increasing to the right and the Y coordinate increasing upwards.

World coordinates denote a common coordinate system across images, generically denoted as a tuple (u, v).

In a particular image, the two coordinate systems are connected by what is called the world coordinate system or WCS. There are many different, complicated forms of WCS, but we never go more complicated than an affine transformation applied to pixel coordinates: (u, v in world coordinates) = A * (x,y in pixel coordinates) + b, where A is a 2x2 (invertible) matrix and b is a 2x1 vector. The values of A and b are encoded in the FITS header in a standard way. Though this is not sufficient to connect pixel coordinates to equatorial coordinates in general, our images always cover such a small piece of the sky that this affine approximation suffices.

We always use world coordinates to denote a unique point in the celestial sphere, in terms of right ascension (sometime denoted by alpha) and declination (sometimes denoted by delta). This is known as the Equatorial coordinate system. So we interpret world coordinates (u, v) as (right ascension, declination), both in degrees, because that's what SDSS uses.

However, a confusing point does arise: SDSS maps right ascension to the Y pixel axis, and declination to the X pixel axis. In other words, the matrix A above looks like

[ 0   dudy;
 dvdx  0  ]

where dudy gives the increment in right ascension for each one-pixel step in the +y direction, and dvdx gives the increment in declination for each one-pixel step in the +x direction.

This means the "vertical" direction in terms of (u, v) world coordinates is not the same as the "vertical" direction in terms of (x, y) pixel coordinates. So be careful.

In order to be consistent with SDSS imagery, we use this convention in all synthetic imagery (see Accuracy benchmarks).