Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 298

computation of grid azimuth, realized by reckoning the grid convergence ʳ and the

arc-to-chord correction of this point.

After the above computations, the curved triangle on a plane constituted by
curved lines will be transformed into the plane triangle formed by straight lines. As
a result, the solution of a triangle and computation of plane coordinates can both be
performed using the formulae for a plane triangle, which can greatly simplify
geodetic computations.
Hence, to reduce the geodetic network on the ellipsoid to the plane, one needs to
carry out computations such as coordinate transformation, arc-to-chord correction,
correction of distance, computation of grid convergence and grid bearing, etc.
Coordinate transformation is one of the direct and inverse solutions of the Gauss
projection, which were introduced in previous sections. Next, our discussions will
focus on other computations.

ʴ

6.5.2 Arc-to-Chord Correction

The correction applied to compensate for the distortion of a straight line when a
geodesic between two points on the ellipsoid is reduced to the chord between
corresponding projection points on the plane is called the arc-to-chord correction,
denoted by

ʴ ij . As the Gauss projection is conformal, the direction of the geodesic
remains unchanged after being projected. Hence, the arc-to-chord correction can
also be interpreted as a process of converting the curve projected from a geodesic to
the chord between the corresponding two points, i.e., the angle between the
projected curve and the chord. Such an angle exists owing to distortions of the
geodesic curves after being projected. Its magnitude depends on the curvature of the
curve, which is therefore also referred to as the curvature correction. It can be seen
that the need for arc-to-chord correction is caused by the fact that, on the plane, the
curves are projected as straight lines, rather than by the distortion of projection.
The accuracy and form of the formulae for the arc-to-chord correction in the
Gauss projection vary with the order of computation. Typically, precise formulae
are applied to the first-order correction, relatively precise formulae are applied to
the second-order net, and for the third- and fourth-order calculations we use the
approximate formulae.

Approximation Formula for Arc-to-Chord Correction

As shown in Fig. 6.12a , the ellipsoid is approximated as a sphere, and the geodesic
P 1 P 2 will be the great circle (orthodrome) on the spherical surface. We draw two
great circles AP 1 and BP 2 passing through points P 1 and P 2 perpendicular to the
central meridian, and both of them intersect the equator at point E. ABP 2 P 1
constitutes a spherical quadrangle. In Fig. 6.12b the geodesic P 1 P 2 is projected as
the curve P 0 P 0 . As Gauss projection is conformal, the great circles AP 1 and BP 2