Sec. 8. Conformal Map Projections

In order to relate the geographic (or geodetic) coordinates of a point on the sphere or spheroid to the rectangular geodetic coor­dinates of the point on a plane the practice is to employ a special method of projecting the surface of the earth’s spheroid onto a projecting plane in what are called zones. The terrestrial globe is thus divided by meridi­ans into six (sometimes three) degree zones extending from the north to the south pole (Fig. 6).

This subdivision of the globe gives 60 zones. These are measured eastward from the zero, Greenwich meridian. Each zone is projected separately onto a cylinder.

If the entire figure of the Earth is considered to be a sphere, the cross section of the cylinder must be a circle. The axis of such a cylin­der AB (Fig. 7) passes through the centre 0 of the sphere and is located in the plane of the earth's equator. Rotating the sphere about its axis, which is perpendicular to that of the cylinder, the sphere's surface is projected in consecutive zones onto the cylindrical surface. Each zone's surface is made to touch the cylinder along the central meridian PP' of the zone.

The representation of the terrain is projected from each zone's spherical surface onto a cylindrical surface using the condition that the angles on the sphere should be equal to those on the cylinder. Once the surfaces of all the zones have been consecutively projected, the cylinder is unfolded onto a plane. We thus obtain a projection of each zone onto a plane surface. This projection is described as a conformal or a conformal transverse cylindrical projection.*(*This projection is termed alternatively the Gauss Projection (or the Gauss Conformal). It is sometimes called the Gauss-Krueger Projection. (The theoretical part is due to Gauss, while Krueger derived the formulae for calculating the projection)).This is because the projection is onto a cylinder whose axis is at right angles to the Earth's axis and the angles of the terrain are not distorted. The zone's cen­tral meridian is represented as a straight line. The entire zone is somewhat expanded when passing from the sphere to a plane.

Referring to Fig. 7, the dashed line shows the boundaries of the zone on the sphere, a solid shows their projection onto the cylinder.

Fig. 6

The lengths of all the lines in this projection are somewhat exagger­ated compared with their natural horizontal projections. The dis­tortions of the lengths are larger, the farther a line is from the zone's central meridian. At the edges of the zones, in latitudes from 30° to 70°, the discrepancies due to distortions in the lengths of the lines in this projection vary from 1 in 1 000 to 1 in 6 000. If such discrep­ancies are impermissible, three-degree zones may be used.

In the Gauss Conformal Projection the distortions of the lines lengths are different at different points of the projection. Yet with respect to directions radiating from a common point the distortions will be uniform. A circle of an infinitesimal radius on the surface of the ellipsoid will be also a circle in this projection. Therefore it is said that this projection preserves infinitesimal figures between a sphere and a projection. Thus, representations of contours of the earth's surface in this projection are very much those which are obtained near the zone's central meridian. In practical engineering operations such distortions may be ignored.

Fig. 7, 8

By projecting successively one zone after another we may represent the entire surface of the globe on the projection (Fig. 8). Such a projection is of no practical importance in geography

because, instead of a continuous representation of the entire earth's surface, it shows ruptures that increase as one approaches the poles. The advantages of this projection are that, first, it enables one to select systems of plane rectangular coordinates over the whole earth's surface with one origin of coordinates for any particular zone. Second, it permits to obtain from the geodetic or geographic coordinate of any point o earth's spheroid or ellipsoid its rectangular geodetic coor­dinates or, conversely, to calculate the corresponding geodetic or geographic coordinates on a sphere or on a spheroid from its plane rectangular coordinates in the projection.