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 12θ. 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.
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| 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 zdμ; 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 = hdz/sin z.
Integrating,
where k is a constant.
Now h is at our disposal and we may give it such a value that two selected parallels are of the correct lengths. Let z1, z2 be the co-latitudes of these parallels, then it is easy to show that
| h = | log sin z1 − log sin z2 | . |
| log tan 12z1 − log tan 12z2 |
This projection, given by equations (i.) and (ii.), is Lambert’s
orthomorphic projection—commonly called Gauss’s projection;
its descriptive name is the orthomorphic conical projection with two
standard parallels.
The constant k in (i.) defines the scale and may be used to render the scale errors along the selected parallels not nil but the same; and some other parallel, e.g. the central parallel may then be made errorless.
The value h = 13, as suggested by Sir John Herschel, is admirably suited for a map of the world. The representation is fan-shaped, with remarkably little distortion (fig. 24).
If any parallel of co-latitude z is true to scale hk(tan 12z1)h = sin z, if this parallel is the equator, so that z1 = 90°, kh = 1, then equation (i.) becomes ρ = (tan 12z)h/h, and the radius of the equator = 1/h. The distance r of any parallel from the equator is 1/h − (tan 12z)h/h = (1/h){1 − (tan 12z)h}.
If, instead of taking the radius of the earth as unity we call it a, r = (a/h){1 − (tan 12z)h}. When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel—that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r = a loge cot 12z, which is the characteristic equation of Mercator’s projection.
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| Fig. 25.—Elliptical equal-area Projection, showing the whole surface of the globe. |
Mercator’s Projection.—From the manner in which we have arrived at this projection it is clear that it retains the characteristic property of orthomorphic projections—namely, similarity of representation of small parts of the surface. In Mercator’s chart the equator is represented by a straight line, which is crossed at right angles by a system of parallel and equidistant straight lines representing the meridians. The parallels are straight lines parallel to the equator, and the distance of the parallel of latitude φ from the equator is, as we have seen above, r = a loge tan (45° + 12φ). In the vicinity of the equator, or indeed within 30° of latitude of the equator, the representation is very accurate, but as we proceed northwards or southwards the exaggeration of area becomes larger, and eventually excessive—the poles being at infinity. This distance of the parallels may be expressed in the form r = a (sin φ + 13 sin3 φ + 15 sin5 φ + . . .), showing that near the equator r is nearly proportional to the latitude. As a consequence of the similar representation of small parts, a curve drawn on the sphere cutting all meridians at the same angle—the loxodromic curve—is projected into a straight line, and it is this property which renders Mercator’s chart so valuable to seamen. For instance: join by a straight line on the chart Land’s End and Bermuda, and measure the angle of intersection of this line with the meridian. We get thus the bearing which a ship has to retain during its course between these ports. This is not great-circle sailing, and the ship so navigated does not take the shortest path. The projection of a great circle (being neither a meridian nor the equator) is a curve which cannot be represented by a simple algebraic equation.
If the true spheroidal shape of the earth is considered, the semiaxes being a and b, putting e = √(a2 − b2) / a, and using common logarithms, the distance of any parallel from the equator can be shown to be
where M, the modulus of common logarithms, = 0.434294. Of course Mercator’s projection was not originally arrived at in the manner above described; the description has been given to show that Mercator’s projection is a particular case of the conical orthomorphic group. The introduction of the projection is due to the fact that for navigation it is very desirable to possess charts which shall give correct local outlines (i.e. in modern phraseology shall be orthomorphic) and shall at the same time show as a straight line any line which cuts the meridians at a constant angle. The latter condition clearly necessitates parallel meridians, and the former a continuous increase of scale as the equator is departed from, i.e. the scale at any point must be equal to the scale at the equator × sec. latitude. In early days the calculations were made by assuming that for a small increase of latitude, say 1′, the scale was constant, then summing up the small lengths so obtained. Nowadays (for simplicity the earth will be taken as a sphere) we should say that a small length of meridian adφ is represented in this projection by a sec φ dφ, and the length of the meridian in the projection between the equator and latitude φ,
which is the direct way of arriving at the law of the construction
