Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 309

Fig. 6.15 Distance
correction in converting
geodesic distance S (left)to
s and D on the Gauss plane
(right)

overlooked for mapping and the use of maps at a scale of 1:5,000 or larger, and the
corresponding corrections, is therefore necessary. One can also use either the 1.5
zone or an urban independent Gauss rectangular coordinate system (i.e., select the
meridian passing through the center of a city as the central meridian) to allow the
distortion of distance to satisfy the mapping needs.

Formula for Distance Correction

Derivations of Formula

In Fig. 6.15 we assume that S is the geodesic distance between two points P 1 and P 2
on the ellipsoid, s is the length of the projected curve between the corresponding
points P 0 1 and P 0 2 projected on the Gauss plane, and D is the chord length between
the two points P 0 1 and P 0 2 on the projected curve.
The correction added while converting the geodesic distance S to the plane chord
D is known as the distance correction, denoted by S.

Generally, the scale factor in a Gauss projection is invariably greater than
1, hence it follows that:

Δ

S < s > D

We are aiming to obtain the relationship between S and D. In the process of
deduction, we will first approach the relationship between S and s, and then that
between s and D. Finally, we will derive the formula for computing the distance
correction

Δ S.