Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 266

ds
dS for (a) an ellipsoid surface and (b) a projection plane

Fig. 6.2 Scale factor m ﾼ

h i

:

2 2

dS 2 r 2

ﾼ ðÞ dq þ ðÞ dL ð 6 : 4 Þ

From ( 6.3 ) we know that q is merely the function of B, which can be called
isometric latitude. The relational expression between the geodetic latitude B and
isometric latitude q can be determined according to:

ð M
r dB,

q ﾼ

which means that, given a geodetic latitude, one corresponding isometric latitude
can be obtained accordingly. Isometric latitude, however, is meaningless in prac-
tical use. There is no reason to calculate the isometric latitude and it is introduced
here only for the convenience of formula derivation.
The arc element ds on the projection plane can be written directly according to
the formula for an arc element on the plane curve:

ds 2

2 2

ﾼ ðÞ dx þ ðÞ dy : ð 6 : 5 Þ

Then we have:

2
ﾼ

2 2

ds
dS

ðÞ dx þ ðÞ dy

m 2

h i :

ﾼ ð 6 : 6 Þ

2 2

r 2 ðÞ dq þ ðÞ dL

The deduction of a conformal condition has to be based on the fact that the scale
factor m is independent of the azimuth A. Thus, the azimuth has to be introduced in
( 6.6 ) to change the above formula. We know that projection means to determine