Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 262

a small area, the Earth's surface can be considered a plane, but for a large area it is
subject to distortion if represented as a plane. Projection is essentially to establish
functional models to allocate such distortions in a reasonable way.
In addition, from Bessel's formula for the solution of the geodetic problem in
Chap. 5 , we know that, although the ellipsoid is a mathematical surface, geodetic
computations carried out on it are complicated. If elements on the ellipsoid are
reduced to a plane, geodetic computations will become much simpler. For instance,
coordinates of points on a plane can be solved according to simple formulae in
plane trigonometry.
Thus, projection from the ellipsoid surface onto a plane is used to control
topographic mapping and to simplify survey computations. The specialized disci-
pline concerning problems of projection is called map projection. In geodetic
survey, the projection of a geodetic control network is also studied in addition to
the projection of coordinates for positions of points.

6.1.2 Definition of Projection

Projection means establishing a one-to-one correspondence between the geodetic
elements on the ellipsoid and the corresponding elements on the plane according to
certain mathematical rules. The former includes geodetic coordinates, geodesic
direction, distance of geodesic line, geodetic azimuth, etc. Once the relationship
between the coordinates of point positions is specified, the corresponding relation-
ship between other elements will be determined as well; hence, the key to deter-
mining the projection relationship is to determine the projection relationship
between the coordinates of point positions.
The “certain mathematical rules” mentioned can be expressed by the following
equations:

x ﾼ F 1 B ð ; L Þ

ð 6 : 1 Þ

y ﾼ F 2 B ð ; L Þ

where (B, L) are the geodetic coordinates of a point on the ellipsoid, and (x, y) are
the rectangular coordinates of this point after being projected onto the plane. It is
obvious that ( 6.1 ) is single-valued, finite, and continuous.
Equation ( 6.1 ) expresses the analytical projection relationship between points on
the ellipsoid and their corresponding points on the projection plane, without any
geometrical meaning. Different projections actually determine the functional forms
of F 1 and F 2 in the formulae according to their specified conditions. The Gauss
projection (and the UTM projection) has its own particular conditions. Once F 1 and
F 2 are determined, the geodetic coordinates of every point on the ellipsoid and the
rectangular coordinates of their respective corresponding points can be determined
accordingly.