Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 302

Fig. 6.13 Arc-to-chord
correction checking

Rearranging gives:

2R m 2
ρ 00

Δʴ 00

Δ P ﾼ

Þ : y m þ ð x 2 x 1

Δʴ 00 ﾼ 0. 1 00 . With y In the third-order triangulation we require that ﾼ 350 km

and x 2 0.1 km. It follows that the approximate

coordinates should be accurate to 0.1 km to meet the desired accuracy in compu-
tations of the third-order arc-to-chord correction. Likewise, for the first- and
second-order arc-to-chord correction, the plane coordinates should be accurate to
10 m and 1 m, respectively. For many third-order triangulations, the desired
accuracy of approximate coordinates is not high and thus iterative computation is
not necessary.

x 1 ﾼ 10 km we get Δ P

Formula for Checking the Computation of Arc-to-Chord Correction

The sum of the measures of the interior angles of an ellipsoidal triangle is 180 ∘ +

,
and that of the triangle formed by curves remains unchanged after the ellipsoidal
triangle is conformally projected onto a plane. As shown in Fig. 6.13 , the curved
triangle A 0 B 0 C 0 is the projection of the ellipsoidal triangle ABC onto the Gauss
plane. Let the angular correction of each angle be

ʵ

ʴ C , which are equal to the
differences in the arc-to-chord correction between the two neighboring sides,
namely:

ʴ A , ʴ B ,

9
=
; :

ʴ A ﾼ ʴ AC ʴ AB
ʴ B ﾼ ʴ BA ʴ BC
ʴ C ﾼ ʴ CB ʴ CA

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