5.4 Projections⁴

A photograph of the sky is the photograph of the inside of a sphere, projected onto a flat surface. In order to accurately map the scheme of co-ordinates used to navigate the sky a projection from sphere to surface must be made.

Cartographers have long been aware of the impossibility of accurately projecting a sphere (e.g., Earth, celestial sphere etc.) on a flat surface (e.g., a map, FITS image etc.). Such a projection cannot be exact in the measurement of surface area and angles at the same time. It may adhere to equivalency, preserving size proportions; or, it may adhere to conformity, thus preserving shapes at the expense of size proportion. The two are mutually exclusive (NYPL, 2001). It is necessary to decide which aspects of the spherical reality are to be rendered on to a flat surface accurately and which aspects will be compromised. These decisions must be made in accordance with the intended use to which the “map” will be put. For optical astronomical work the decision is made by the telescope optics as to how the celestial sphere (or part thereof) is projected on to a plane surface.

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The image surface will be a plane orthogonal to the optical axis of the telescope and tangential to the celestial sphere (see Figure 5.1). Calabretta (1992) summaries: “Zenithal (also known as azimuthal) projections are a class of projection in which the surface of projection is a plane, the native co-ordinate system is such that the polar axis is orthogonal to the plane of projection at the reference point” (see Figure 5.2, left panel), “whence the meridians are projected as equi-spaced rays emanating from a central point, and the parallels are mapped as concentric circles centred on the same point”. All zenithal projections are constructed with the pole of the native co-ordinate system at the reference point, therefore,

(5.1)

with the projection being completely defined by R and . Perspective zenithal projections are generated from a point and carried through the unit sphere to the plane of projection (see Figure 5.3, left panel). The Field of view of most optical telescopes is close to a zenithal gnomonic projection (Calabretta and Greisen, 2000). This projection is classified by the fact that; the projection point is at a distance = 0 from the centre of the unit sphere (see Figure 5.3, right panel).

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The gnomonic projection is a geometric projection of the celestial sphere onto a tangent plane with = 0. It is illustrated in Figure 5.4, the projection is neither conformal (angle preserving) nor equivalent (equal area) but does have the property that all straight lines in the projection are great circles on the sphere, this is because the the projection is from the centre of the sphere. The derivation of this projection is performed by Greisen (1994) (see also Calabretta and Greisen, 2001).