| ρ = | 2.034 sin u | . |
| 1.47 + cos u |
For a map of Africa or South America, the limiting radius β we may take as 40°; then in this case
| ρ = | 2.543 sin u | . |
| 1.625 + cos u |
For Asia, β = 54, and the distance h of the point of sight in this case is 1.61. Fig. 11 is a map of Asia having the meridians and parallels laid down on this system.
| Fig. 11. |
Fig. 12 is a perspective representation of more than a hemisphere, the radius β being 108°, and the distance h of the point of vision, 1.40.
| Fig. 12.—Twilight Projection. Clarke’s Perspective Projection for a Spherical Radius of 108°. |
The co-ordinates xy of any point in this perspective may be expressed in terms of latitude and longitude of the corresponding point on the sphere in the following manner. The co-ordinates originating at the centre take the central meridian for the axis of y and a line perpendicular to it for the axis of x. Let the latitude of the point G, which is to occupy the centre of the map, be γ; if φ, ω be the latitude and longitude of any point P (the longitude being reckoned from the meridian of G), u the distance PG, and μ the azimuth of P at G, then the spherical triangle whose sides are 90° − γ, 90° − φ, and u gives these relations—
| sin u sin μ = cos φ sin ω, |
| sin u cos μ = cos γ sin φ − sin γ cos φ cos ω, |
| cos u = sin γ sin φ + cos γ cos φ cos ω. |
Now x = ρ sin μ, y = ρ cos μ, that is,