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SECTION 8. WORLD COORDINATE SYSTEMS

Table 8.1.

WCS and Celestial Coordinates Notation

Variable(s)

Meaning

Related FITS Keywords

i

Index variable for world coordinates

···

j

Index variable for pixel coordinates

···

a

Alternative WCS version code

···

pj

Pixel coordinates

···

rj

Reference pixel coordinates

CRPIXj a

mij

Linear transformation matrix

CDi_ja or PCi_ja

si

Coordinate scales

CDELTi a

(x, y)

Projection plane coordinates

···

(φ, θ)

Native longitude and latitude

···

(α, δ)

Celestial longitude and latitude

···

PVi_1a  † PVi_2a  †

(φ0, θ0)

Native longitude and latitude of the fiducial point

,

(α0, δ0)

Celestial longitude and latitude of the fiducial point

CRVALi a

(αp, δp)

Celestial longitude and latitude of the native pole

···

LONPOLEa (=PVi_3a  †,

(φp, θp)

Native longitude and latitude of the celestial pole

)

LATPOLEa (=PVi_4a  †

)

†

Associated with longitude axis i.

of various reserved keywords with elements of a transformation (or a series of trans-

formations), or with parameters of a projection function. The conversion from pixel

coordinates in the data array to world coordinates is simply a matter of applying the

specified transformations (in order) via the appropriate keyword values; conversely,

defining a WCS for an image amounts to solving for the elements of the transformation

matrix(es) or coefficients of the function(s) of interest and recording them in the form

of WCS keyword values. The description of the WCS systems and their expression in

FITS HDUs is quite extensive and detailed, but is aided by a careful choice of notation.

Key elements of the notation are summarized in Table 8.1, and are used throughout this

section. The formal definitions of the keywords appear in the following subsections.

The conversion of image pixel coordinates to world coordinates is a multi-step pro-

cess, as illustrated in Figure 8.1.

For all coordinate types, the first step is a linear transformation applied via matrix

multiplication of the vector of pixel coordinate elements, pj :

qi =

Summation over j = 1 to N of mij (pj - rj )

(8.1)

where rj are the pixel coordinate elements of the reference point, j indexes the pixel

FITS Standard