General Theory of Zenithal Projections.—For the sake of simplicity it will be at first assumed that the pole is the centre of the map, and that the earth is a sphere. According to what has been said above, the meridians are now straight lines diverging from the pole, dividing the 360° into equal angles; and the parallels are represented by circles having the pole as centre, the radius of the parallel whose co-latitude is z being ρ, a certain function of z. The particular function selected determines the nature of the projection.
| Fig. 23. |
Let Ppq, Prs (fig. 23) be two contiguous meridians crossed by parallels rp, sq, and Op′q′, Or′s′ the straight lines representing these meridians. If the angle at P is dμ, this also is the value of the angle at O. Let the co-latitude
Pp = z, Pq = z + dz; Op′ = ρ, Oq′ = ρ + dρ,
the circular arcs p′r′, q′s′ representing the parallels pr, qs. If the radius of the sphere be unity,
p′q′ = dρ; p′r′ = ρ dμ,
pq = dz; pr = sin z dμ.
Put
σ = dρ / dz; σ′ = ρ / sin z,
then p′q′ = σpq and p′r′ = σ′pr. That is to say, σ, σ′ may be regarded as the relative scales, at co-latitude z, of the representation, σ applying to meridional measurements, σ′ to measurements perpendicular to the meridian. A small square situated in co-latitude z, having one side in the direction of the meridian—the length of its side being i—is represented by a rectangle whose sides are iσ and iσ′; its area consequently is i2σσ′.
If it were possible to make a perfect representation, then we should have σ = 1, σ′ = 1 throughout. This, however, is impossible. We may make σ = 1 throughout by taking ρ = z. This is the Equidistant Projection just described, a very simple and effective method of representation.