For simplicity of explanation we have supposed this method of development so applied as to have the pole in the centre. There is, however, no necessity for this, and any point on the surface of the sphere may be taken as the centre. All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. Then the azimuth is represented unaltered, and any spherical distance z is represented by ρ. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation.
Thus treating the earth as a sphere and applying the Zenithal Equal-area Projection to the case of Africa, the central point selected being on the equator, we have, if θ be the spherical distance of any point from the centre, φ, α the latitude and longitude (with reference to the centre), of this point, cos θ = cos φ cos α. If A is the azimuth of this point at the centre, tan A = sin α cot φ. On paper a line from the centre is drawn at an azimuth A, and the distance θ is represented by 2 sin 1⁄2θ. This makes a very good projection for a single-sheet equal-area map of Africa. The exaggeration in such systems, it is important to remember, whether of linear scale, area, or angle, is the same for a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the distance from the centre only.
General Theory of Conical Projections.
| Fig. 24. |
Meridians are represented by straight lines drawn through a point, and a difference of longitude ω is represented by an angle hω. The parallels of latitude are circular arcs, all having as centre the point of divergence of the meridian lines. It is clear that perspective and zenithal projections are particular groups of conical projections.
Let z be the co-latitude of a parallel, and ρ, a function of z, the radius of the circle representing this parallel. Consider the infinitely small space on the sphere contained by two consecutive meridians, the difference of whose longitude is dμ, and two consecutive parallels whose co-latitudes are z and z + dz. The sides of this rectangle are pq = dz, pr = sin z dμ; in the projection p′q′r′s′ these become p′q′ = dρ, and p′r′ = ρhdμ.
The scales of the projection as compared with the sphere are p′q′/pq = dρ/dz = the scale of meridian measurements = σ, say, and p′r′/pr = ρhdμ/sin zdμ = ρh/sin z = scale of measurements perpendicular to the meridian = σ′, say.
Now we may make σ = 1 throughout, then ρ = z + const. This gives either the group of conical projections with rectified meridians, or as a particular case the equidistant zenithal.
We may make σ = σ′ throughout, which is the same as requiring that at any point the scale shall be the same in all directions. This gives a group of orthomorphic projections.
In this case dρ/dz = ρh/sin z, or dρ/ρ = h dz/sin z.