Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 313

Thus, it can be seen that with y ﾼ 0 and Δ S ﾼ 0, the distance correction on the

central meridian is zero. In the case of y 0, we see that the distance

correction is always a positive value, increasing with the distance from the central
meridian. When y m ﾼ

6ﾼ 0 and Δ S >

300 km, S ﾼ 5 km, and R m ﾼ 6, 400 km, we can get Δ S

ﾼ 6 m after the distance correction. Obviously, the distance correction is

non-negligible in computations of all orders.
Also, if converting the planar chord-length D to the geodesic distance S,we
have:

S ﾼ D Δ S ð 6 : 75 Þ

Accuracy of Coordinates Required in the Computation of Distance
Correction

To calculate the distance correction one needs to know the plane coordinates of a
point. As the value of the distance correction is not large, it does not require too
high a level of coordinate accuracy. To know the approximations of coordinates is
sufficient; we will analyze the desired degree of accuracy for coordinates.
From ( 6.74 ), we get:

2y m

Δ ð D S Þﾼ

2R m 2 S

Δ y

and

R m 2
y m S Δ

Δ y ﾼ ð D S Þ:

With y m ﾼ 350 km, S ﾼ 50 km, and R m ﾼ 6, 400 km, and for the first-order

triangulation, let Δ (D S) ﾼ 0.001 m; we obtain Δ y ﾼ 2.34 m. In the meantime,

for the second- and third-order triangulations, y is 23.4 m and 234 m, respectively.

Hence, the coordinates with an accuracy of 1 m and 10 m would satisfy the first-,
second-, and third-order triangulations. In a large number of computations, to avoid
accumulative errors of coordinates, the coordinates are always accurate to 0.1 m,
1 m, and 10 m. Based on the analyses above, such an accuracy of coordinates can
also satisfy the computation requirements for the arc-to-chord correction of the
same order.

Δ