particular value of h which corresponds to the required maximum can be obtained by interpolation. Thus we find that if it be required to make the best possible perspective representation of a hemisphere, the values of h and k are h = 1.47 and k = 2.034; so that in this case
| 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.
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| 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.
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| 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,
| 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 | |
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| 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.
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| (From Text Book of Topographical Surveying, by permission of the Controller of H. M. Stationery Office.) |
| Fig. 14.—Part of the Atlantic Ocean on a Meridian Central Projection. The shortest path between any two points is shown on this projection by a straight line. |
Fig. 14 is an example of a meridian central projection of part of the Atlantic Ocean. The term “gnomonic” was applied
