Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 303

The interior angles of an ellipsoidal triangle add up to 180 ∘ + . After angular

corrections are applied, the ellipsoidal triangle becomes a triangle formed by
straight line sides with the sum of interior angles 180 ∘ , namely:

ʵ

180 ∘ þ ʵ þ ʴ A þ ʴ B þ ʴ C 180 ∘ :

ð Þ ﾼ

Then the spherical excess is:

ʵ ﾼʴ A þ ʴ B þ ʴ C ð Þ ð 6 : 60 Þ

The above expression shows that the spherical excess of each triangle is equal to
the sum of the angular corrections of the interior angles of this plane triangle, of
opposite signs. Equation ( 6.60 ) can be used to check the correctness of the arc-to-
chord correction and computations of the spherical excess.

Practical Formulae

The formulae accurate to 0.001 00 are:

2 0 1 3 9

=
;

1:2 ﾼ ρ 00

y m 3
R m 2

6 ʷ m 2 t m

R m

00

4 @ A þ 5

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

6R m 2

: ð 6 : 61 Þ

ρ 00
6R m 2

00
2:1

ʴ ﾼʴ 00 1:2 þ ð x 2 x 1 Þ ð y 2 y 1 Þ

For instance, given that x 1 ﾼ 3602547.8 m, y 1 ﾼ 298960.0 m, x 2 ﾼ 3584223.0 m,

32 ∘ 25.5 0 , ʴ 00 1.2 ﾼ + 14.294 00 ,and

y 2 ﾼ 323655.4 m and B m ﾼ the results give

00 ﾼ

14.678 00 . ʴ 2.1

We check:

P ʴ ﾼ ʴ 1 þʴ 2 þʴ 3

123 00 , ﾼ ð 16 : 430 } 14 : 294 } Þþ ð 14 : 678 } 2 : 577 } Þþ ð 2 : 523 }þ 16 : 519 } Þ ﾼ 1 :

+ 1.123 00 .
The two results have the same absolute value, indicating that the computation is
correct, as shown in Fig. 6.14 . Owing to the round-off errors, there would presum-
ably be a difference of 0.001 "-0.002 " between them. After being proved errorless,
the value of difference can be assigned to the large angle.

and ʵ ﾼ