ZENITHAL PROJECTIONS Some point on the earth is se lected 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.
If is the co-latitude of the centre of the map, z the co latitude of any other point, a 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 0= tan z cos a, then cos c= cos z sec 0 cos (z —0), and sin A = sin z sin a 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 'equi distant 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 z) —1. General Theory of Zenithal Projections.—For the sake of simplicity it will be at first as sumed that the pole is the centre of the map, and that the earth is a sphere. The meridians are now straight lines diverging from the pole, and the parallels are rep resented by circles having the pole as centre, the radius of the parallel whose co-latitude is z being p, a function of z determining the nature of the projection.
Let Ppq, Prs (fig. 3o) 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 d,u, this also is the value of the angle at 0. Let
then p'q'= apq and p'r'=a'pr. That is to say, r, a' may be re garded as the relative scales, at co-latitude z, of the representa tion, a applying to meridional measurements, a' 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 iv and ia'; its area consequently is Ara'. If it were pos sible to make a perfect representation, then we should have = I, a' = I throughout. This, however, is impossible.
We may make a = r throughout by taking p = z. This is the Equidistant Projection, a very simple and effective method of representation.
If we make a'= i throughout this gives p = sin z, a perspec tive projection, namely, the Orthographic.
We may require that areas be strictly represented in the development. This will be effected by making or pdp = sin zdz, the integral of which is p= 2 sinlz, which is the Zenithal Equal-area Projection of Lambert, sometimes wrongly referred to as Lorgna's Projection after Antonio Lorgna (b. 1736). In this system there is misrepresentation of form, but no mis representation of areas.
We may require a projection in which all small parts are to be represented in their true forms, i.e., an orthomorphic projection. This condition will be attained by making =a', or dp/ p= dz/sin z, the integral of which is, c being an arbitrary constant, p = c tan z. This, again, is a perspective projection, namely, the Stereographic. In this the whole is not a similar representation of the original. The scale, a= 2 at any point, applies to all directions round that point.
These two last projections are, as it were, at the extremes of the scale. We may avoid both extremes by the following considerations. Although we cannot make i and = i, so as to have a perfect picture of the spherical surface, yet consider ing a — I and a' — i as the local errors of the representation, we may make (a — + (a' a minimum over the whole surface to be represented. This projection, known as the Projection by Balance of Errors, is due to Sir George Airy (q.v.). Airy's in vestigation was corrected by Clarke (Phil. Mag. 1862), and A. E. Young has extended the principle to the conical class generally (see Some Investigations in the Theory of Map Projections, i920) .