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2.1.2. Linear transformation matrix

The proposal to replace the CROTAi keywords of Wells et al. ([1981]) with a general linear transformation matrix dates from Hanisch & Wells ([1988]), although the details of its implementation have undergone considerable evolution. The main point of divergence has been whether the matrix should completely replace or simply augment the CDELTi, but there are also important differences relating to the default values of the matrix elements.

In defining a nomenclature which augments the CDELTi we have been guided by the following considerations:

In the PCi_j formalism, the matrix elements mij are encoded in

PC i_j (floating value)

header cards, and si as CDELTi. The i and j indices are used without leading zeroes, e.g. PC1_1 and CDELT1. The default values for PCi_j are 1.0 for i = j and 0.0 otherwise. The PCi_j matrix must not be singular; it must have an inverse. Furthermore, all CDELTi must be non-zero. In other words, invertibility means that transformations which project from an initial coordinate system of dimensionality WCSAXES to a world coordinate system of dimensionality less than WCSAXES are forbidden.

In the CDi_j formalism Eqs. (1) and (2) are combined as

Equation 3 (3)

and the

CD i_j (floating-valued)

keywords encode the product simij. The i and j indices are used without leading zeroes, e.g. CD1_1. The CD i_j matrix must not be singular; it must have an inverse. CDELTi and CROTAi are allowed to coexist with CDi_j as an aid to old FITS interpreters, but are to be ignored by new readers. The default behavior for CDi_j differs from that for PCi_j; if one or more CDi_j cards are present then all unspecified CDi_j default to zero. If no CDi_j cards are present then the header is assumed to be in PCi_j form whether or not any PCi_j cards are present since this results in an interpretation of CDELTi consistent with Wells et al. ([1981]).

We specifically prohibit mixing of the PCi_j and CDi_j nomenclatures in any FITS header data unit. With this restriction, translation from the CDi_j formalism to the PCi_j formalism is effected simply in the keyword parsing stage of header interpretation; the CDi_j should be considered equivalent to the PCi_j subject to the considerations for default values noted above and with CDELTi set to unity. Similarly, CDi_j can be calculated from PCi_j and CDELTi following Eq. 3.

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