let tan [theta] = tan z cos [alpha], then cos c = cos z sec [theta]
cos (z - [theta]), and sin A = sin z sin [alpha] 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 [rho], a certain function of z. The particular
function selected determines the nature of the projection.
[Illustration: 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 d[mu], this also is the value of
the angle at O. Let the co-latitude
Pp = z, Pq = z + dz; Op´ = [rho], Oq´ = [rho] + d[rho],
the circular arcs p´r´, q´s´ representing the parallels pr, qs. If the
radius of the sphere be unity,
p´q´ = d[rho]; p´r´ = [rho] d[mu],
pq = dz; pr = sin z d[mu].