Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 316

l 3

1
3 sin B cos 2 B 1

t 2

2 4

tan ʳ ﾼ sin B l þ þ þ 3 ʷ þ 2 ʷ

l 5

1
15 sin B cos 4 B 2

4t 2 2t 4 þ þ þ :

With tan ʳ ﾼ x

tan 1 x

1

3 x 3

1

5 x 5

1

3 tan 3

1

5 tan 5

and ʳ ﾼ ﾼ x þ þﾼ tan ʳ ʳ þ ʳ þ

the result is:

" #

:

l 00 2 cos 2 B
3

l 00 4 cos 4 B
15

þ

ʳ 00 ﾼ l 00 sin B 1

2 4

t 2 þ 1 þ 3 ʷ þ 2 ʷ 2 ð 6 : 78 Þ

ρ 00 2 ρ 00 4

Equation ( 6.78 ) is the formula for computing the grid convergence ʳ from

geodetic coordinates (L, B). It can thus be seen that:

1. When l 0, i.e., the grid convergence is zero on

both the central meridian and the equator.
2. Grid convergence

ﾼ 0, ʳ ﾼ 0 and when B ﾼ 0, ʳ ﾼ

are both considered to be
positive when point P is east of the central meridian, and negative when P is west
of the central meridian.
3. When the latitude B is constant, the value of

ʳ is the odd function of l. l and ʳ

increases with the increasing
difference in longitude between the central meridian and point P.
4. When l is constant, the value of

ʳ

ʳ increases as latitude increases towards the

poles. becomes greatest at the poles.

Equation ( 6.78 ) can be accurate to 0.001 00 when l

ʳ

3.5 o . wWhen l 2 , the

term that contains l 5 is less than 0.001 00 and can be neglected.

To Compute Given Plane Coordinates (x, y)

γ

The formula for computing the grid convergence from given plane coordinates x,
y can be obtained by making changes to ( 6.78 ), in which we replace l by Cartesian
coordinates and B by B f . We will derive the formula to the term that contains y 3 .
B is replaced by B f through expanding sinB using the Taylor series:

ﾼ

sin B ﾼ sin B f B f B sin B f cos B f B f B ,

where (B f B) is obtained by taking the principal terms in the first expression of

( 6.46 ):

ﾼ

t f t f

2N f 2M f N f y 2

y 2

2
f

B f B ﾼ 1 þ ʷ :