square of unit side occupying the same position, when transformed
to a rectangle, has its sides 1.02 and 1.15, its area 1.17,
and its diagonals intersect at 90° ± 7° 6′. The latter projection
is therefore the best in point of “similarity,” but the former
represents areas best. This applies, however, only to a particular
part of the map; along the equator towards 30° or 40°
longitude, the polyconic is certainly inferior, while along the
meridian it is better than the perspective—except, of course,
near the centre. Upon the whole the more even distribution
of distortion gives the advantage to the perspective system.
For single sheets on large scales there is nothing to choose
between this projection and the simple polyconic. Both are
sensibly perfect representations. The rectangular polyconic
is occasionally used by the topographical section of the general
staff.
Zenithal Projections.
Some point on the earth is selected as the central point of the map; great circles radiating from this point are represented by straight lines which are inclined at their true angles at the point of intersection. Distances along the radiating lines vary according to any law outwards from the centre. It follows (on the spherical assumption), that circles of which the selected point is the centre are also circles on the projection. It is obvious that all perspective projections are zenithal.
Equidistant Zenithal Projection.—In this projection, which is commonly called the “equidistant projection,” any point on the sphere being taken as the centre of the map, great circles through this point are represented by straight lines of the true rectified lengths, and intersect each other at the true angles.
In the general case—
if z1 is the co-latitude of the centre of the map, z the co-latitude of any other point, α the difference of longitude of the two points, A the azimuth of the line joining them, and c the spherical length of the line joining them, then the position of the intersection of any meridian with any parallel is given (on the spherical assumption) by the solution of a simple spherical triangle.
Thus—
let tan θ = tan z cos α, then cos c = cos z sec θ cos (z − θ), and sin A = sin z sin α cosec c.
The most useful case is that in which the central point is the pole; the meridians are straight lines inclined to each other at the true angular differences of longitude, and the parallels are equidistant circles with the pole as centre. This is the best projection to use for maps exhibiting the progress of polar discovery, and is called the polar equidistant projection. The errors are smaller than might be supposed. There are no scale errors along the meridians, and along the parallels the scale error is (z / sin x) − 1, where z is the co-latitude of the parallel. On a parallel 10° distant from the pole the error of scale is only 0.5%.
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
the circular arcs p′r ′, q′s′ representing the parallels pr, qs. If the radius of the sphere be unity,
pq = dz; pr = sin z dμ.
Put
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.
Or we may make σ′= 1 throughout. This gives ρ = sin z, a perspective projection, namely, the Orthographic.
Or we may require that areas be strictly represented in the development. This will be effected by making σσ′ = 1, or ρ dρ = sin zdz, the integral of which is ρ = 2 sin 12z, which is the Zenithal Equal-area Projection of Lambert, sometimes, though wrongly referred to as Lorgna’s Projection after Antonio Lorgna (b. 1736). In this system there is misrepresentation of form, but no misrepresentation of areas.
Or we may require a projection in which all small parts are to be represented in their true forms i.e. an orthomorphic projection. For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be attained by making σ = σ′, or dρ/ρ = dz/sin z, the integral of which is, c being an arbitrary constant, ρ = c tan 12z. This, again, is a perspective projection, namely, the Stereographic. In this, though all small parts of the surface are represented in their correct shapes, yet, the scale varying from one part of the map to another, the whole is not a similar representation of the original. The scale, σ = 12c sec2 12z, at any point, applies to all directions round that point.
These two last projections are, as it were, at the extremes of the scale; each, perfect in its own way, is in other respects objectionable. We may avoid both extremes by the following considerations. Although we cannot make σ = 1 and σ′ = 1, so as to have a perfect picture of the spherical surface, yet considering σ − 1 and σ′ − 1 as the local errors of the representation, we may make (σ − 1)2 + (σ′ − 1)2 a minimum over the whole surface to be represented. To effect this we must multiply this expression by the element of surface to which it applies, viz. sin zd zd μ, and then integrate from the centre to the (circular) limits of the map. Let β be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as
| ∫β0 { ( | dρ | − 1 )2 + ( | ρ | − 1 )2 } sin zdz, |
| dz | sin z |
which is to be made a minimum. Putting ρ = z + y, and giving to y only a variation subject to the condition δy = 0 when z = 0, the equations of solution—using the ordinary notation of the calculus of variations—are
| N − | d(P) | = 0; Pβ = 0, |
| dz |
Pβ being the value of 2p sin z when z = β. This gives
| sin2 z | d2y | + sin z cos z | dy | − y = z − sin z ( | dy | ) β = 0. |
| dz2 | dz | dz |
This method of development is due to Sir George Airy, whose original paper—the investigation is different in form from the above, which is due to Colonel Clarke—will be found in the Philosophical Magazine for 1861. The solution of the differential equation leads to this result—
| ρ = 2 cot 12z loge sec 12z + C tan 12z, |
| C = 2 cot2 12β loge sec 12β. |
The limiting radius of the map is R = 2C tan 12β. In this system, called by Sir George Airy Projection by balance of errors, the total misrepresentation is an absolute minimum. For short it may be called Airy’s Projection.
Returning to the general case where ρ is any function of z, let us consider the local misrepresentation of direction. Take any indefinitely small line, length = i, making an angle α with the meridian in co-latitude z. Its projections on a meridian and parallel are i cos α, i sin α, which in the map are represented by iσ cos α, iσ′ sin α. If then α′ be the angle in the map corresponding to α,
Put
and the error α′ − α of representation = ε, then
| tan ε = | (Σ − 1) tan α | . |
| 1 + Σ tan2 α |
Put Σ = cot2 ζ, then ε is a maximum when α = ζ, and the corresponding value of ε is
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
