Map Projections

Fig. 16 shows an orthographic projec tion of the sphere on the horizon of any place.

Stereographic Projection.

In this case the point of vision is on the surface, and the projection is made on the plane of the great circle whose pole is V. Let kplV (fig. 17) be a great circle through the point of vision, and ors the trace of the plane of projection. Let c be the centre of a small circle whose radius is cp=c/.

Since the representation of every infinitely small circle on the surface is itself a circle, it follows that in' this projection the repre sentation of small parts is strictly similar. Another inference is that the angle in which two lines on the sphere intersect is repre sented by the same angle in the projection. In this projection the angles are correctly represented and every small triangle is repre sented by a similar triangle. Projections having this property are called orthomorphic or conformable.

External Perspective Projection.

We now come to the general case in which the point of vision has any position outside the sphere. Let abed (fig. 19) be the great circle section of the sphere by a plane passing through c, the central point of the por tion of surface to be represented, and V the point of vision. Let pj perpendicular to 1'c be the plane of representation, join mV cutting pj in f, then f is the projection of any point m in the circle abc, and of is the representation of cm (see Phil. Mag. 1862).

Clarke's Projection.—The constants h and k can be determined, so that the total misrepresentation, viz. : I — 02+ — 021 sin /du, shall be a minimum, being the great est value of it, or the spherical radius of the map. Fig. 20 is a map of Asia having the meridians and parallels laid down on this system.

Central or Gnomonic (Perspective) Projection.

In this projection the eye is imagined to be at the centre of the sphere. It is evident that, since the planes of all great circles of the sphere pass through the centre, the representations of all great circles on this projection will be straight lines, and this is the special property of the central projection, that any great circle (i.e., short

est line on the spherical surface) is represented by a straight line. The plane of projection may be either parallel to the plane of the equator, in which case the parallels are represented by concentric circles and the meridians by straight lines radiating from the common centre; it may be parallel to the plane of some meridian, the meridians being parallel straight lines and the parallels hyper bolas; or it may be inclined to the axis of the sphere at any angle X, in which case the central meridian is a straight line at right angles to the equator, which is also a straight line (fig. so). These three varieties of the central projection are known as polar, meridian or horizontal.

Fig. 22 is an example of a meridian central projection of part of the Atlantic ocean. The term gnomonic was applied to this projection because the projection of the meridians is a similar problem to that of the gradua tion of a sun-dial. The gnomonic 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 engen dered by a study of maps on Mercator's projection, may be cor rected by an inspection of maps drawn on the central projection.

Until 1922 it was thought that the gnomonic was the only projection which showed the great circles as straight lines, but H. Maurer has shown that, if the projection be compressed uni formly in any direction, the great circles will still be represented as straight lines. He calls this new projection orthodronzic.

Moreover, there are two points in this projection which have the property that azimuths from them to any other points are cor rect. This gives a means for plot ting ships' positions from bear ings sent by wireless from two points (see Zeitschrift fiir Ver messungswesen, 1922.