to this projection because the projection of the meridians is a
similar problem to that of the graduation of a sun-dial. It is,
however, better to use the
term “central,” which
explains itself. The central
projection is useful
for the study of direct
routes by sea and land.
The United States Hydrographic
Department has
published some charts on
this projection. False
notions of the direction
of shortest lines, which
are engendered by a study
of maps on Mercator’s
projection, may be corrected
by an inspection
of maps drawn on the
central projection.
There is no projection which accurately possesses the property of showing shortest paths by straight lines when applied to the spheroid; one which very nearly does so is that which results from the intersection of terrestrial normals with a plane.
We have briefly reviewed the most important projections which are derived from the sphere by direct geometrical construction, and we pass to that more important branch of the subject which deals with projections which are not subject to this limitation.
Conical Projections.
Conical projections are those in which the parallels are represented by concentric circles and the meridians by equally spaced radii. There is no necessary connexion between a conical projection and any touching or secant cone. Projections for instance which are derived by geometrical construction from secant cones are very poor projections, exhibiting large errors, and they will not be discussed. The name conical is given to the group embraced by the above definition, because, as is obvious, a projection so drawn can be bent round to form a cone. The simplest and, at the same time, one of the most useful forms of conical projection is the following:
 |
| Fig. 15. |
Conical Projection with Rectified Meridians and Two Standard Parallels.—In some books this has been, most unfortunately, termed the “secant conical,” on account of the fact that there are two parallels of the correct length. The use of this term in the past has caused much confusion. Two selected parallels are represented by concentric circular arcs of their true lengths; the meridians are their radii. The degrees along the meridians are represented by their true lengths; and the other parallels are circular arcs through points so determined and are concentric with the chosen parallels.
Thus in fig. 15 two parallels Gn and G′n′ are represented by their true lengths on the sphere; all the distances along the meridian PGG′, pnn′ are the true spherical lengths rectified.
Let γ be the co-latitude of Gn; γ′ that of Gn′; ω be the true difference of longitude of PGG′ and pnn′; hω be the angle at O; and OP = z, where Pp is the representation of the pole. Then the true length of parallel Gn on the sphere is ω sin γ, and this is equal to the length on the projection, i.e. ω sin γ = hω (z + γ); similarly ω sin γ′ = hω (z + γ′).
The radius of the sphere is assumed to be unity, and z and γ are expressed in circular measure. Hence h = sin γ/(z + γ) = sin γ′ (z + γ′); from this h and z are easily found.
In the above description it has been assumed that the two errorless parallels have been selected. But it is usually desirable to impose some condition which itself will fix the errorless parallels. There are many conditions, any one of which may be imposed. In fig. 15 let Cm and C′m′ represent the extreme parallels of the map, and let the co-latitudes of these parallels be c and c′, then any one of the following conditions may be fulfilled:—
(a) The errors of scale of the extreme parallels may be made equal and may be equated to the error of scale of the parallel of maximum error (which is near the mean parallel).
(b) Or the errors of scale of the extreme parallels may be equated to that of the mean parallel. This is not so good a projection as (a).
(c) Or the absolute errors of the extreme and mean parallels may be equated.
(d) Or in the last the parallel of maximum error may be considered instead of the mean parallel.
(e) Or the mean length of all the parallels may be made correct. This is equivalent to making the total area between the extreme parallels correct, and must be combined with another condition, for example, that the errors of scale on the extreme parallels shall be equal.
We will now discuss (a) above, viz. a conical projection with rectified meridians and two standard parallels, the scale errors of the extreme parallels and parallel of maximum error being equated.
Since the scale errors of the extreme parallels are to be equal,
| h (z + c) | − 1 = | h (z + c′) | − 1, whence z = | c′ sin c − c sin c′ | . |
| sin c | sin c′ | sin c′ − sin c |
The error of scale along any parallel (near the centre), of which the co-latitude is b is
This is a maximum when
Also
| 1 − | h (z + b) | = | h (z + c) | − 1 whence h is found. |
| sin b | sin c |
For the errorless parallels of co-latitudes γ and γ′ we have
If this is applied to the case of a map of South Africa between the limits 15° S. and 35° S. (see fig. 16) it will be found that the parallel of maximum error is 25° 20′; the errorless parallels, to the nearest degree, are those of 18° and 32°. The greatest scale error in this case is about 0.7%.
In the above account the earth has been treated as a sphere. Of course its real shape is approximately a spheroid of revolution, and the values of the axes most commonly employed are those of Clarke or of Bessel. For the spheroid, formulae arrived at by the same principles but more cumbrous in shape must be used. But it will usually be sufficient for the selection of the errorless parallels to use the simple spherical formulae given above; then, having made the selection of these parallels, the true spheroidal lengths along the meridians between them can be taken out of the ordinary tables (such as those published by the Ordnance Survey or by the U.S. Coast and Geodetic Survey). Thus, if a1, a2, are the lengths of 1° of the errorless parallels (taken from the tables), d the true rectified length of the meridian arc between them (taken from the tables),
and the radius on paper of parallel, a1 is a1d/(a2 − a1), and the radius of any other parallel = radius of a1 ± the true meridian distance between the parallels.
This class of projection was used for the 1/1,000,000 Ordnance map of the British Isles. The three maximum scale errors in this case work out to 0.23%, the range of the projection being from 50° N. to 61° N., and the errorless parallels are 59° 31′ and 51°44′.
Where no great refinement is required it will be sufficient to take the errorless parallels as those distant from the extreme parallels about one-sixth of the total range in latitude. Thus suppose it is required to plot a projection for India between latitudes 8° and 40° N. By this rough rule the errorless parallels should be distant from the extreme parallels about 32°/6, i.e. 5° 20′; they should therefore, to the nearest degree, be 13° and 35° N. The maximum scale errors will be about 2%.
The scale errors vary approximately as the square of the range of latitude; a rough rule is, largest scale error = L2/50,000, where L is the range in the latitude in degrees. Thus a country with a range of 7° in latitude (nearly 500 m.) can be plotted on this projection with a maximum linear scale error (along a parallel) of about 0.1%;[1] there is no error along any meridian. It is immaterial with this
- ↑ This error is much less than that which may be expected from contraction and expansion of the paper upon which the projection is drawn or printed.
