8.1. BASIC CONCEPTS

75

Pixel

Coordinates

CRPIXj,

Linear transformation:

PCi_j or

translation, rotation,

skew, scale

CDi_j

Intermediate Pixel

Coordinates

Rescale to

CDELTi

physical units

Intermediate World

Coordinates

CTYPEi,

Coordinate

projection, offset

CRVALi

PVi_m

World

Coordinates

Figure 8.1 A schematic view of converting pixel coordinates to world coordinates.

axis, and i the world axis. The mij matrix is a non-singular, square matrix of dimension

N ×N , where N is the number of world coordinate axes. The elements qi of the resulting

intermediate pixel coordinate vector are offsets, in dimensionless pixel units, from the

reference point along axes coincident with those of the intermediate world coordinates.

Thus, the conversion of qi to the corresponding intermediate world coordinate element

xi is a simple scale:

xi = siqi.

(8.2)

There are three conventions for associating FITS keywords with the above trans-

formations. In the first formalism, the matrix elements mij are encoded in the PCi j

keywords and the scale factors si are encoded in the CDELTi keywords, which must have

non-zero values. In the second formalism Eqs. (8.1) and (8.2) are combined as

xi =

summation over j = 1 to N of (simij )(pj - rj )

(8.3)

and the CDi j keywords encode the product simij . The third convention was widely used

before the development of the 2 previously described conventions and uses the CDELTi

keywords to define the image scale and the CROTA2 keyword to define a bulk rotation of

FITS Standard