Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 301

The relatively precise formulae for the arc-to-chord correction are:

=

1:2 ﾼ ρ 00

00

ʴ ð x 2 x 1 Þ ð 2y 1 þ y 2 Þ

6R m 2

: ð 6 : 57 Þ

ρ 00
6R m 2

;

00
2

ʴ ﾼ ð x 2 x 1 Þ ð 2y 2 þ y 1 Þ

: 1

The average side-length of China's second-order triangulation network is 13 km.
When y m is less than 250 km, the above equations are accurate to 0.01 " and are
typically used in computations of second-order triangulation. When y m is greater
than 250 km, we should apply the precise formulae for arc-to-chord correction in
( 6.58 ):

0
@

1
A ρ 00 ʷ m 2 t m
R m

9
=

1:2 ﾼ ρ 00

y m 3
R m 2

00

y m 2 ʴ ð x 2 x 1 Þ 2y 1 þ y 2 ð y 2 y 1 Þ

6R m 2

0
@

1
A þ ρ 00 ʷ m 2 t m
R m

: ð 6 : 58 Þ

;

ρ 00
6R m 2

y m 3
R m 2

00
2:1

y m 2

ʴ ﾼ ð x 2 x 1 Þ 2y 2 þ y 1 ð y 2 y 1 Þ

The above formulae are accurate to 0.001 00 and are applicable to computations of
the first-order triangulation.

Accuracy of Coordinates Required in Computations of Arc-to-Chord
Correction

To calculate the arc-to-chord correction, we should first obtain the plane coordi-
nates of a point. Paradoxically, knowing precisely the plane coordinates of a point
also requires that the arc-to-chord correction be computed first. The way to resolve
this contradiction is to apply the iterative computing method. As computations of
different orders require different degrees of accuracy, the number of iterations is
also different. Here we will analyze the accuracy of coordinates required.
Taking the total differential of ( 6.56 ) gives:

ρ 00
2R m 2

Δʴ 00 ﾼ

ﾽ y m Δ ð x 2 x 1 Þþ ð x 2 x 1 Þ Δ y :

We set Δ (x 2 x 1 ) ﾼ Δ y ﾼ Δ P, to obtain:

y m þ

ρ 00
2R m 2 Δ

Δʴ 00 ﾼ

P ð x 2 x 1 Þ: