Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 305

2
þ

2

∂ x ∂ y

G
r 2 ﾼ

∂ l ∂ l

m 2 ﾼ : ð 6 : 62 Þ

N 2 cos 2 B

Using the Geodetic Coordinates (B, l ) to Derive the Formula for m

We take the partial derivative with respect to l by applying the formula for the direct
solution of the Gauss projection ( 6.28 ) to get:

9
=

l 3

∂ x

∂

N
6 sin Bcos 3 B 5

t 2

2 4

l ﾼ

Nsin Bcos Bl þ þ 9 ʷ þ 4 ʷ

120 sin Bcos 5 B 61

t 4 l 5

N

58t 2 þ þ

: ð 6 : 63 Þ

;

l 2

∂ y

∂

N
2 cos 3 B 1

t 2

2

l ﾼ

N cos B þ þʷ

24 cos 5 B 518t 2

2 t 2 l 4

N

t 4

2

þ þ þ14ʷ 58ʷ

l and ∂ y

In ( 6.63 ), we divide the obtained ∂ x
∂ l (with the terms containing l 5 and

2 l 3

ʷ

∂

left out) by N cos B; then the square is:

0
@

1
A

2

∂ x

∂ l

l 4
3 sin 2 B cos 2 B 5

l 2 sin 2 B t 2

N 2 cos 2 B ﾼ

þ

0 1

2

∂ y

@ A

2 3

2

∂ l

l 2
2 cos 2 B 1

þ

l 4
24 cos 4 B 5

4 5

t 2

2

18t 2 t 4

N 2 cos 2 B ﾼ

1 þ þ ʷ þ

l 4
4 cos 4 B 1

2

l 4
12 cos 4 B 5

l 2 cos 2 B 1 t 2 18t 2 t 4

t 2

2

ﾼ 1 þ ð þ ʷ Þ þ þ þ

l 4
3 cos 4 B 2

:

l 2 cos 2 B 1 t 2

2

6t 2 t 4

ﾼ 1 þ ð þ ʷ Þ þ þ