Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 315

(i.e., P 0 N 0 and P 0 E 0 ) are still perpendicular to each other. Hence, the angular
difference between P 0 E 0 and the horizontal grid line P 0 R is also the grid conver-
gence . The grid convergence is used to convert the geodetic azimuth and grid

azimuth, and vice versa. It can be calculated given either the geodetic coordinates
(L, B) or grid coordinates (x, y). The derivations of their formulae are provided
below.

ʳ

To Compute γ Given Geodetic Coordinates (L, B)

From Fig. 6.17 , on the projected parallel P 0 , according to the geometric meaning of
first-order derivative, we get:

dx
dy :

tan ʳ ﾼ

Taking the total differential of x ﾼ f 1 (q, l ), y ﾼ f 2 (q, l ) yields:

ﾼ ∂

x

þ ∂

x
∂

dx

q dq l dl, ∂

ﾼ ∂

y

þ ∂

y

dy

q dq l dl

:

∂ ∂

On the parallel curve P 0 E 0 , B is constant, and therefore in this case dq

ﾼ 0. Hence,

we have:

9
=

ﾼ ∂

x
∂

dx

l dl

,

ﾼ ∂

y
∂

;

dy

l dl

so

∂ x

∂ l

tan ʳ ﾼ : ð 6 : 77 Þ

∂ y

∂ l

From ( 6.63 ) we have obtained the partial derivatives of x and y with respect to l.
Inserting into the above equation and rearranging produces: