Geodetic Projections

representations of the surface of the terrestrial ellipsoid on a plane surface, done according to specific rules. Geodetic projections are used in numerical work with a geodetic grid and for the solution of various practical problems involving the results of in-place geodetic measurement, as well as for construction of topographical maps on scales greater than 1:1,000,000. The theory of geodetic projections has much in common with the theory of cartographic projections, but in a cartographic projection a minimum of distortion is required above all, whereas geodetic projections require that such distortions be strictly and simply taken into consideration. Use of the points of a geodetic grid as base points in surveying a locality makes it necessary to fit survey materials into this grid without any further adjustment to the plane surface, apart from making reductions in scale. This necessity determines the choice of geodetic projections; they have to come from among those conformal projections which have the property that at every point of the projection a constant scale is maintained in all directions within the limits of a small area which has this point as its center. That is to say, the similarity of the original and its representation is to be maintained. If the coordinates of reference points of a survey are computed very accurately in the geodetic projection selected, the scale will thereby be taken into account automatically, and no further reductions of survey materials will be required. The manner in which the surface of the ellipsoid is divided into parts (zones) depends on the geodetic projection chosen. In the theory of geodetic projections, formulas are given that make it possible to transfer the coordinates of points and the lengths and directions of lines in an exact manner from an ellipsoid onto a plane surface (and vice versa), to compute scale, and to pass from one zone of a projection to another. When such an analytical apparatus is available and when calculations relative to the reference point of a geodetic grid and its initial side are performed, one can then view this grid on the plane surface of the geodetic projection and perform operations based on plane trigonometry and analytical geometry.

Geodetic projections include the Gauss-Krüger projection, the Lambert conic conformal projection, and various types of stereographic projection, among others. The Gauss-Krüger projection is used in the USSR and a number of other countries. It is defined as a conformal projection of an ellipsoid onto a plane surface in which there are no distortions on the axial meridian, represented by a straight line which is the projection’s axis of symmetry. The surface of the ellipsoid in this case is divided by meridians into coordinate zones which extend from one pole to the other. The width of the longitudinal zones is set at 6° and 3°. In each zone the representation of the axial meridian is taken as the axis of the abscissa, and the representation of the equator is taken as the axis of the ordinate.