2.1. Coordinate definition and computation
In the current proposal, we regard the conversion of pixel coordinates to world coordinates as a multi-step process. This is illustrated conceptually in Fig. 1, which shows only the steps to be discussed here. Later extensions may interpose additional steps as required. For example, Paper II divides the final step into two with the computation of intermediate spherical coordinates that are subsequently converted to celestial coordinates via a spherical rotation. Paper IV interposes optional distortion corrections between the first and/or second steps of Fig. 1. Generally these are intended to account for small residuals that cannot be described by one of the standard world coordinate transformations. These may arise in a variety of ways; naturally (e.g. aberration or atmospheric refraction), via complex instrumental response functions (e.g. data cubes produced by a Fabry-Perot interferometer for which surfaces of constant wavelength are curved), by the intrinsic nature of the system under study (e.g. surface coordinates of the asteroid Eros), or as a result of instrumental peculiarities.
Figure 1. Conversion of pixel coordinates to world coordinates shown as a multi-step process. In the first step a linear transformation is applied via matrix multiplication of the pixel coordinate vector. This linear transformation may be restricted to the geometrical effects of rotation and skewness with scaling to physical units deferred until the second step (PCi_j plus CDELTi formalism). Alternatively, scaling may be applied via the matrix with the second step omitted (CDi_j formalism). The final step applies a possibly non-linear transformation to produce the final world coordinates. Although generic keywords for this step are defined in this paper, the mathematical details, including the interpretation of the INTERMEDIATE WORLD COORDINATES, are deferred to later papers which may also interpose additional steps in the algorithm chain. For later reference, the mathematical symbols associated with each step are shown in the box at right.