Editing Gnomonic projection (section)

==Properties==
The gnomonic projection is from the centre of a sphere to a plane tangent to the sphere (Fig 1 below). The sphere and the plane touch at the tangent point. Great circles transform to straight lines via the gnomonic projection. Since [[Meridian (geography)|meridians]] (lines of longitude) and the [[equator]] are great circles, they are always shown as straight lines on a gnomonic map. Since the projection is from the centre of the sphere, a gnomonic map can represent less than half of the area of the sphere. Distortion of the scale of the map increases from the centre (tangent point) to the periphery.<ref name=Snyder/>

*If the tangent point is one of the [[Geographical pole|poles]] then the meridians  are radial and equally spaced (Fig 2 below). The equator cannot be shown as it is at [[infinity]] in all directions. Other [[Circle of latitude|parallels]] (lines of latitude) are depicted as concentric [[circle]]s.

*If the tangent point is on the equator then the meridians are parallel but not equally spaced (Fig 3 below). The equator is a straight line perpendicular to the meridians. Other parallels are depicted as [[hyperbola]]e.

*If the tangent point is not on a pole or the equator, then the meridians are radially outward straight lines from a pole, but not equally spaced (Fig 4 below). The equator is a straight line that is perpendicular to only one meridian, indicating that the projection is not [[conformal map|conformal]]. Other parallels are depicted as [[conic section]]s.

{{Gallery
|width=200
|align=left
|footer=Figs 2 - 4 are from Snyder (1987) Figure 34{{r|Snyder|p=166}}.
|File:Gnomonic.png 
|Fig 1. A great circle projects to a straight line in the gnomonic projection
|File:Snyder Figure 34 Gnomonic A.jpg
|Fig 2. Gnomonic projection centered on the north pole
|File:Snyder Figure 34 Gnomonic B.jpg
|Fig 3. Gnomonic projection centered on the equator
|File:Snyder Figure 34 Gnomonic C.jpg
|Fig 4. Gnomonic projection centered on latitude 40° north
}}
{{clear}}


As with all [[azimuth]]al projections, angles from the tangent point are preserved. The map distance from that point is a function ''r''(''d'') of the [[great-circle distance|true distance]] ''d'', given by

:<math> r(d) = R\,\tan \frac d R</math>

where ''R'' is the radius of the Earth. The radial scale is

:<math> r'(d) = \frac{1}{\cos^2\frac d R} </math>

and the [[:wikt:transverse|transverse]] scale

: <math> \frac{1}{\cos\frac d R} </math>

so the transverse scale increases outwardly, and the radial scale even more.