Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 288

B ﾼ Fq ðÞ: ð 6 : 39 Þ

Similarly, we have:

Fq f :

B f ﾼ ð 6 : 40 Þ

, and according to the Taylor series we

We expand B ﾼ Fq ðÞﾼ Fq f þ q q f

obtain:

Fq f þ

þ

d 2 B
dq 2

2

dB
dq

1
2

B ﾼ q q f q q f þ ... ð 6 : 41 Þ

!

f f

It follows from ( 6.3 ) that:

=
;

0 1 0 1 0 1

dB
dq

r
M

N cos B
M

@ A f ﾼ @ A f ﾼ @ A f ﾼ

V f 2 cos B f

0 1

: ð 6 : 42 Þ

d 2 B
dq 2

@ A f ﾼ

ʷ f 2

cos B f sin B f 1 þ 4

Inserting ( 6.40 ) and ( 6.42 ) into ( 6.41 ) gives:

q 2

1
2 cos B f sin B f 1

V f 2 cos B f q ʷ f 2

B ﾼ B f þ q f þ 4 q f : ð 6 : 43 Þ

According to the first equation in ( 6.38 ), we get:

9
=

y 4

ﾼ

t f sec B f
2N f

t sec B f
24N f

y 2 6t f þ ʷ

2
f

q q f þ 5 þ

: ð 6 : 44 Þ

t f sec 2 B f
4N f

2

;

y 4

q q f ﾼ

Inserting again into ( 6.43 ) and rearranging yields:

y 4

t f t f

24M f N f

2M f N f y 2

3t f þ ʷ

2 2

f t f

B ﾼ B f þ 5 þ

f

9 ʷ : ð 6 : 45 Þ

Combining ( 6.45 ) with the second equation in ( 6.38 ), the formula for the inverse
solution of the Gauss projection can be obtained as follows: