General Theory of Conical Projections

GENERAL THEORY OF CONICAL PROJECTIONS Meridians are represented by straight lines drawn through a point, and a difference of longitude w is represented by an angle hw. The parallels of latitude are circular arcs, all having as centre the point of divergence of the meridian lines.

Let z be the co-latitude of a parallel, and p, a function of z, the radius of the circle representing this parallel. Consider the infinitely small space on the sphere contained by two con secutive meridians, the differ ence of whose longitude is du, and two consecutive parallels whose co-latitudes are z and z-Edz (fig. 31). The sides of this rectangle are pq = dz, pr = sin ; in the projection p'q'r's' these become p'q' =dp, and p'r' = phdµ.

The scales of the projection as compared with the sphere are p'q7pq =dp/dz= the scale of meridian measurements = a, say, and p'r'/pr = phdi.t/ sin zd,u= ph/sin z = scale of measurements perpendicular to the meridian =a', say.

Now we may make a =I throughout, then p=z+const. This gives either the group of conical projections with rectified meridians, or as a particular case the equidistant zenithal.

We may make a =a' 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 dp/dz= ph/sin z, or dp/p=hdz/sin z.

Integrating,

p=k(tan lz)h, (i.) 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 be the co-latitudes of these parallels, then it can be shown that log sin z, —log sin h = log tan —log tan 1z2 (ii.) 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.

Mercator's Projection.

This projection retains the charac teristic property of orthomorphic projections, viz., similarity of representation of small parts of the surface. The equator is represented by a straight line, which is crossed at right angles at equidistant points by a system of lines representing the me ridians. The parallels are straight lines parallel to the equator, and the distance of the parallel of latitude (1) from the equator is, as we have seen above, r =a tan (45°-1-10). In the vicinity of the equator, or indeed within 30° of latitude of the equator, the representation is good, but as we proceed northwards or southwards the exaggeration of area becomes larger, and eventu ally excessive—the poles being at infinity. This distance of the

parallels may be expressed in the form r = a (sin OH- sin '0+1 sin .), showing that near the equator r is nearly propor tional to the latitude. As a consequence of the similar representa tion of small parts, a curve drawn on the sphere cutting all me ridians at the same angle—the loxodromic curve—is projected into a straight line, and it is this property which renders Mer cator's chart so valuable to seamen. The projection of a great circle (being neither a meridian nor the equator) is a curve which cannot be represented by a simple algebraic equa tion. Mercator's projection although indispensable at sea, is of little value for land maps. The misconceptions arising from this exaggeration of scale may, however, be corrected by the juxta position of a map of the world on an equal-area projection.

It is now necessary to revert to the general consideration of conical projections. It has been shown that the scales of the projection (fig. 20) as compared with the sphere are p'q'/pq =dp/dz=o- along a meridian, and p'r'/ =ph/sin z = e at right angles to a meridian. Now if o-e= r the areas are correctly rep resented, then hpdp = sin zdz, and integrating z; (i.)this gives the whole group of equal-area conical projections. As a special case let the pole be the centre of the projected parallel, then when z =0, p = 0, and const =1, we have p= 2 sin 15/Sh (ii.)Let be the co-latitude of some parallel which is to be correctly represented, then 2h sin =sin and h = cos' putting this value of h in equation (ii.) the radius of any parallel =p= 2 sin lz sec This is Lambert's conical equal-area projection with one stand ard parallel, the pole being the centre of the parallels.

If we put e then h=i, and the meridians are inclined at their true angles, also the scale at the pole becomes correct, and equation (iii.) becomes p= 2 sin lz; (iv.) this is the zenithal equal-area projection.

Reverting to the general expression for equal-area conical projections, p=v{2(c—cos z)/h} (i.) we can dispose of C and h so that any two selected parallels shall be their true lengths; let their co-latitudes be and then 2h(C—cos (v.) 52) = (vi.) from which C and h are easily found, and the radii are obtained from (i.). This is H. C. Albers' conical equal-area projection with two standard parallels. The pole is not the centre of the parallels.