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The following two sections (5.4 & 5.5) are concerned with the theory of mapping a spherical image, or part thereof onto a plane, and how to integrate this work with the FITS specification. The large majority of this work has been carried out by Calabretta and Greisen. The work started in circa 1992, with the most complete form of the work to date being 3 papers written by these authors. However the formalism laid down for integrating a WCS with FITS has never been accepted by the FITS committees and therefore the papers have never been published in any journals, however they have been repeatedly presented at several conferences. The work is widely accepted by the astronomical community at large and is the implementation used by most observatories producing FITS files with WCSs, including that used by the ING WFS. The 3 papers are continually updated and are available at the home pages of the authors, the most recent changes occurred in late 2001 with several FITS keywords being changed. This occurred well after the completion of the LT PL and so has not been integrated into the PL. However the mapping and projection theory stays the same. The versions of the papers used in the PL, and explained in this chapter, for the FITS implementation work are Calabretta and Greisen (2000); Greisen (2000); Greisen and Calabretta (2000), their location is specified in the bibliography.