Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 306

Substituting into ( 6.62 ) yields:

l 4
3 cos 4 B 2

6t 2 t 4

m 2 l 2 cos 2 B 1 t 2

2

l 2 sin 2 B

ﾼ 1 þ ð þ ʷ Þ þ þ þ

l 4
3 sin 2 B cos 2 B 5

t 2

þ

l 4
3 cos 4 B 2

l 2 cos 2 B 1 t 2

ﾼ 1 þ ð þ ʷ

2

Þþ

1=2

l 4

l 2 cos 2 B 1

2

3 cos 4 B 2

t 2 m ﾼ 1 þ ð þ ʷ Þþ ð Þ :

The above is the approximate formula used to compute the scale factor m from
given geodetic coordinates (B, l ). If one more terms in the equation are allowed for
in-formula derivations, we will then be able to achieve a formula for the scale
m with a higher level of accuracy. Expanding the above equation according to the
binomial theorem 1 þ

1 = 2

1 1

8 x 2

ð x Þ ﾼ 1 þ

2 x

þ ...gives:

2 3

þ

l 4
3 cos 4 B 2

l 2
8

2

1
2

4 5

l 2 cos 2 B 1

2

t 2 cos 2 B 1

2

m ﾼ 1 þ þ ʷ þ ʷ

þ

l 2
2 cos 2 B 1

l 4
6 cos 4 B 2

l 4
8 cos 4 B

2

t 2 ﾼ 1 þ þ ʷ

l 2
2 cos 2 B 1

þ

l 4
24 cos 4 B 5

:

2

4t 2 ﾼ 1 þ þ ʷ

If l is measured in seconds, then we have:

l 00 2
2

l 00 4
24

þ ,

ρ 00 2 cos 2 B 1

2

ρ 00 4 cos 4 B 5

4t 2

m ﾼ 1 þ þ ʷ ð 6 : 64 Þ

which is the formula for the scale factor expressed by geodetic coordinates.

To Derive m Given Gauss Plane Coordinates (x, y)

According to the second equation in ( 6.28 ), and leaving out the m 5 term, we obtain:

l 3

l 2
6 cos 2 B 1

N
6 cos 3 B 1

t 2

2

t 2

2

y ﾼ N cos Bl þ þ ʷ ﾼ Nl cos B 1 þ þ ʷ :