CONICAL PROJECTIONS Conical projections are those in which the parallels are repre sented by concentric circles and the meridians by equally spaced radii. There is no necessary connection between a conical pro jection and any touching or secant cone. Projections which are derived by geometrical construc tion from secant cones exhibit large errors, and will not be dis cussed.
Conical Projection with Rectified Meridians and Two Standard Parallels.—In some books this has been termed the "secant conical," the use of which term has caused much con fusion. 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. 23 two parallels Gn and G'n' are represented by their true lengths on the sphere; all the distances along the meridians PGG', pnn' are the true. spherical lengths rectified. Let be the co-latitude of Gn; 'y' that of Gn'; w be the true difference of longitude of PGG' and pnn'; hco be the angle at 0; and OP= z, where Pp is the representation of the pole. Then the true length of parallel Gn on the sphere is co sin 7, and this is equal to the length on the projection. The radius of the sphere is assumed to be unity, and z and y are expressed in circular meas ure, hence h and z are easily found. 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 error less parallels. In fig. 23 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 ful filled : (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). This class of projection is ac curate, simple and useful.
(b) Or the errors of scale of the extreme parallels may be equated to that of the mean parallel.
(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 cor rect. 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.
Imagine a cone to touch the sphere along any selected parallel, the radius of this parallel on paper (Pp, fig. 25) will be r cot 4), where r is the radius of the sphere and 4 is the latitude; or if the spheroidal shape is taken into account, the radius of the parallel on paper will be v cot 4) where v is the normal terminated by the minor axis (the value v can be found from ordi nary geodetic tables). The meridians are generators of the cone and every paral lel such as HH' is a circle, concentric with the selected parallel Pp and distant from it the true rectified length of the meridian arc between them.
The errors of scale along the parallels increase rapidly as the selected parallel is departed from, the parallels on paper being always too large. The projection has no merits as compared with the group just described.
Bonne's Projection is derived from the simple conical in the following way: a central meridian is chosen and drawn as a straight line; degrees of latitude spaced at the true rectified distances are marked along this line; the parallels are concentric circular arcs drawn through the proper points on the central meridian, the centre of the arcs being fixed by describing one chosen par allel with a radius of v cot 4); the me ridians on each side of the central meridian are drawn through distances at.the true lengths along the paral lels on sphere or spheroid (fig. 26).
This system is ill-adapted for countries having great extent in longitude. Where an equal-area projection is required for a country such as France, Scotland or Madagascar, this projec tion is a good one.
Sinusoidal Equal-area Projection, sometimes known as Sanson's and sometimes incorrectly called Flamsteed's, is a particular case of Bonne's in which the selected parallel is the equator. It is a very suitable projection for an equal-area map of Africa (fig. 27).