Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 290

where B f is the footprint latitude; namely, the corresponding geodetic latitude with
x X (the length of the meridian arc from the equator, where X corresponds to X f ).

Footprint latitude B f can be obtained by iteration. Equation (5.41) that corresponds
to the Krassowski Ellipsoid is taken as an example to illustrate.
Given X, to compute B f reversely, when iteration starts, the initial value is set:

ﾼ

B ðÞ
f

ﾼ X = 111134 : 8611 : ð 6 : 50 Þ

The ensuing iterative procedure follows:

f

= B iþ1

ð Þ

FB ðÞ ﾼ X 111134 : 8611, ð 6 : 51 Þ

f

f

32005

FB ðÞ 7799 sin B ðÞ

f

9238 sin 3 B ðÞ
f

6973 sin 5 B ðÞ
f

ﾼ : þ 133 : þ 0 :

cos B ðÞ
f

0039 sin 7 B ðÞ
f

þ 0 : : ð 6 : 52 Þ

B i f 10 8 , to ensure that B f is

accurate to 0.0001 00 . Normally, iterating five times will achieve the desired accu-
racy. It should be noted that in programming computation, B f obtained using the
iteration formula is measured in “degrees,” whereas in iterative calculations, B f in
the trigonometric function is measured in “radians,” so one needs to convert in
iterative procedures.
Similarly, for GRS75 or GRS80 Ellipsoids, given X to compute B f reservedly,
one can iterate using (5.42) and (5.43), respectively.

B f (i +1)

Iterations are repeated until j j < 1

Formulae for the Inverse Solution of the Gauss Projection (Accurate to 0.01 00 ;
The Results Are Measured in Degrees)

It follows from ( 6.46 ) that:

9
=

2 0 1 0 1 3

2 4

1
2 V f t f

y
N f

1
12

y
N f

180
ˀ

4 @ A @ A 5

B ∘ ﾼ B f 3t f þʷ

2 2

f t f

5 þ

f

9 ʷ

,

2
4

0
@

1
A

0
@

1
A

0
@

1
A

3
5

;

3 5

1
cosB f

y
N f

1
6

y
N f

1
120

y
N f

180
ˀ

l ∘ ﾼ 2t f þʷ

2
f

28t f þ 24t f 1 þ þ 5 þ

ð 6 : 53 Þ

where B f can be obtained by iteration. The method is as stated above. The number of
decimal places can be reduced taking into consideration the circumstances.