1 Introduction

This paper is the second in a series which establishes conventions by which world coordinates may be associated with FITS (Hanisch et al. 2001) image, random groups, and table data. Paper I (Greisen & Calabretta 2002) lays the groundwork by developing general constructs and related FITS header keywords and the rules for their usage in recording coordinate information. In Paper III, Greisen et al. (2002) apply these methods to spectral coordinates. Paper IV (Calabretta et al. 2002) extends the formalism to deal with general distortions of the coordinate grid. This paper, Paper II, addresses the specific problem of describing celestial coordinates in a two-dimensional projection of the sky. As such it generalizes the informal but widely used conventions established by Greisen (1983, 1986) for the Astronomical Image Processing System, hereinafter referred to as the AIPS convention.

Paper I describes the computation of world coordinates as a multi-step process. Pixel coordinates are linearly transformed to intermediate world coordinates that in the final step are transformed into the required world coordinates.

Figure 1: Conversion of pixel coordinates to celestial coordinates. The INTERMEDIATE WORLD COORDINATES of Paper I, Fig. 1 are here interpreted as PROJECTION PLANE COORDINATES, i.e. Cartesian coordinates in the plane of projection, and the multiple steps required to produce them have been condensed into one. This paper is concerned in particular with the steps enclosed in the dotted box. For later reference, the mathematical symbols associated with each step are shown in the box at right (see also Tables 1 and 13).

In this paper we associate particular elements of the intermediate world coordinates with Cartesian coordinates in the plane of the spherical projection. Figure 1, by analogy with Fig. 1 of Paper I, focuses on the transformation as it applies to these projection plane coordinates. The final step is here divided into two sub-steps, a spherical projection defined in terms of a convenient coordinate system which we refer to as native spherical coordinates, followed by a spherical rotation of these native coordinates to the required celestial coordinate system.

The original FITS paper by Wells et al. (1981) introduced the CRPIX ja keyword to define the pixel coordinates $(r_1,r_2,r_3,\ldots)$ of a coordinate reference point. Paper I retains this but replaces the coordinate rotation keywords CROTA i with a linear transformation matrix. Thus, the transformation of pixel coordinates $(p_1,p_2,p_3,\ldots)$ to intermediate world coordinates $(x_1,x_2,x_3,\ldots)$ becomes

 

$$s_i \sum_{j=1}^{N} m_{ij} (p_j - r_j) ,$$ (1)

$$\hphantom{s_i} \sum_{j=1}^{N} (s_i m_{ij}) (p_j - r_j) ,$$

where N is the number of axes given by the NAXIS keyword. As suggested by the two forms of the equation, the scales s_i, and matrix elements m_ij may be represented either separately or in combination. In the first form s_i is given by CDELT ia and m_ij by PC i_ja; in the second, the product s_i m_ij is given by CD i_ja. The two forms may not coexist in one coordinate representation.

Equation (1) establishes that the reference point is the origin of intermediate world coordinates. We require that the linear transformation be constructed so that the plane of projection is defined by two axes of the x_i coordinate space. We will refer to intermediate world coordinates in this plane as projection plane coordinates, (x,y), thus with reference point at (x,y) = (0,0). Note that this does not necessarily correspond to any plane defined by the p_j axes since the linear transformation matrix may introduce rotation and/or skew.

Wells et al. (1981) established that all angles in FITS were to be measured in degrees and this has been entrenched by the AIPS convention and confirmed in the IAU-endorsed FITS standard (Hanisch et al. 2001). Paper I introduced the CUNIT ia keyword to define the units of CRVAL ia and CDELT ia. Accordingly, we require CUNIT ia = 'deg' for the celestial CRVAL ia and CDELT ia, whether given explicitly or not. Consequently, the (x,y) coordinates in the plane of projection are measured in degrees. For consistency, we use degree measure for native and celestial spherical coordinates and for all other angular measures in this paper.

For linear coordinate systems Wells et al. (1981) prescribed that world coordinates should be computed simply by adding the relative world coordinates, x_i, to the coordinate value at the reference point given by CRVAL ia. Paper I extends this by providing that particular values of CTYPE ia may be established by convention to denote non-linear transformations involving predefined functions of the x_i parameterized by the CRVAL ia keyword values and possibly other parameters.

In Sects. 2, 3 and 5 of this paper we will define the functions for the transformation from (x,y)coordinates in the plane of projection to celestial spherical coordinates for all spherical map projections likely to be of use in astronomy.

The FITS header keywords discussed within the main body of this paper apply to the primary image header and image extension headers. Image fragments within binary tables extensions defined by Cotton et al. (1995) have additional nomenclature requirements, a solution for which was proposed in Paper I. Coordinate descriptions may also be associated with the random groups data format defined by Greisen & Harten (1981). These issues will be expanded upon in Sect. 4.

Section 6 considers the translation of older AIPS convention FITS headers to the new system and provisions that may be made to support older FITS-reading programs. Section 7 discusses the concepts presented here, including worked examples of header interpretation and construction.