x= cos φ sin ω,
k h + sin γ sin φ + cos γ cos φ cos ω
y= cos γ sin φ − sin γ cos φ cos ω,
k h + sin γ sin φ + cos γ cos φ cos ω

by which x and y can be computed for any point of the sphere. If from these equations we eliminate ω, we get the equation to the parallel whose latitude is φ; it is an ellipse whose centre is in the central meridian, and its greater axis perpendicular to the same. The radius of curvature of this ellipse at its intersection with the centre meridian is k cos φ / (h sin γ + sin φ).

The elimination of φ between x and y gives the equation of the meridian whose longitude is ω, which also is an ellipse whose centre and axes may be determined.

The following table contains the computed co-ordinates for a map of Africa, which is included between latitudes 40° north and 40° south and 40° of longitude east and west of a central meridian.

φ Values of x and y.
ω = 0° ω = 10° ω = 20° ω = 30° ω = 40°
 0° x =  0.00  9.69 19.43 29.25 39.17
y =  0.00  0.00  0.00  0.00  0.00
10° x =  0.00  9.60 19.24 28.95 38.76
y =  9.69  9.75  9.92 10.21 10.63
20° x =  0.00  9.32 18.67 28.07 37.53
y = 19.43 19.54 19.87 20.43 21.25
30° x =  0.00  8.84 17.70 26.56 35.44
y = 29.25 29.40 29.87 30.67 31.83
40° x =  0.00  8.15 16.28 24.39 32.44
y = 39.17 39.36 39.94 40.93 42.34
Fig. 13.

Central or Gnomonic (Perspective) Projection.—In this projection the eye is imagined to be at the centre of the sphere. It is evident that, since the planes of all great circles of the sphere pass through the centre, the representations of all great circles on this projection will be straight lines, and this is the special property of the central projection, that any great circle (i.e. shortest line on the spherical surface) is represented by a straight line. The plane of projection may be either parallel to the plane of the equator, in which case the parallels are represented by concentric circles and the meridians by straight lines radiating from the common centre; or the plane of projection may be parallel to the plane of some meridian, in which case the meridians are parallel straight lines and the parallels are hyperbolas; or the plane of projection may be inclined to the axis of the sphere at any angle λ.

In the latter case, which is the most general, if θ is the angle any meridian makes (on paper) with the central meridian, α the longitude of any point P with reference to the central meridian, l the latitude of P, then it is clear that the central meridian is a straight line at right angles to the equator, which is also a straight line, also tan θ = sin λ tan α, and the distance of p, the projection of P, from the equator along its meridian is (on paper) m sec α sin l / sin (l + x), where tan x = cot λ cos α, and m is a constant which defines the scale.

The three varieties of the central projection are, as is the case with other perspective projections, known as polar, meridian or horizontal, according to the inclination of the plane of projection.