Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 280

c
V cos B
c
V 3

2 0 1 3

dn 1
dq ﾼ

c
V 2

1
V ʷ

c
V sin B

4 @ A cos B 5

2 t

2
4

3
5 V 2 cos B

2
4

3
5 V 2 cos B

c
V 3

c
V 3

2

V 2 ﾼ sin B ʷ ﾼ sin B

c
V sin B cos B

ﾼ ﾼ N sin B cos B :

Inserting into the third equation in ( 6.22 ) gives:

N
2 sin B cos B

m 2 ﾼ : ð 6 : 25 Þ

Taking derivatives in turn from m 2, and substituting correspondingly into ( 6.22 ),
we get n 3 , m 4 , n 5 ... as:

9
=

N
6 cos 3 B 1

t 2

2

n 3 ﾼ þ ʷ

N
24 sin B cos 3 B 5

t 2

2

m 4 ﾼ þ 9 ʷ

: ð 6 : 26 Þ

;

N
120 cos 5 B 5

18t 2 t 4

n 5 ﾼ þ

⋮

2 l 5
and l 6 or higher orders, produces the formula for the direct solution of the Gauss
projection:

Substituting ( 6.24 ), ( 6.25 ), and ( 6.26 ) into ( 6.23 ) and neglecting terms like ʷ

9
=
;

l 00 4

N N

24 ρ 00 2 sinB cos Bl 00 2 ρ 00 4 sinB cos 3 B 5

t 2

2

x ﾼ X þ þ þ 9 ʷ

2

,

l 00 3

ρ 00 5 cos 5 B 5 18t 2

t 4 l 00 5

N
ρ 00

N N

120

cos Bl 00 þ

ρ 00 3 cos 3 B 1

t 2

2

y ﾼ þ ʷ þ þ

6

ð 6 : 27 Þ

206264.80625 00 . l 00 (unit: second) is the difference between the longi-
tude of point P on the ellipsoid and that of the central meridian. l is positive when
point P is east of the central meridian, and negative when P is west of the central
meridian. B is the geodetic latitude of P. X is the length of the meridian arc from the
equator to point of latitude B. Given the geodetic coordinates (L, B) of point P (the

ρ 00 ﾼ where