Gauss and UTM Conformal Projections and the Plane Rectangular Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 268

2 2

ds 2 ﾼ E dq ðÞ þ 2F dq ðÞ ðÞþ dl G dl ðÞ : ð 6 : 9 Þ

Substituting the above equation into ( 6.6 ) produces:

2 2

E dq ðÞ þ 2F dq ðÞ ðÞþ dl G dl ðÞ

m 2 ﾼ h i : ð6:10Þ

2 2

r 2

ðÞ dq þ ðÞ dl

Equation ( 6.10 ) does not include the direction-dependent elements. To introduce
the condition that “scale factor m is independent of azimuth A”, we need to further
convert ( 6.10 ).
From Fig. 6.2 , we have:

P 1 P 2
PP 2 ﾼ

MdB
rdl ﾼ

dq
dl

tan 90 ∘ ð A Þ ﾼ ,

and consequently:

dl ﾼ tan Adq : ð6:11Þ

Inserting the above equation into ( 6.10 ) gives:

2 2 2

E dq ðÞ þ 2F tan A dq ðÞ þ G tan 2 A dq ðÞ E þ 2F tan A G tan 2 A

r 2 sec 2 A

þ

m 2

h i

ﾼ ﾼ

2 2

r 2 ðÞ dq þ tan 2 A dq ðÞ

E cos 2 A G sin 2 A þ 2F sin A cos A þ

ﾼ : ð 6 : 12 Þ

r 2

6.2.3 General Condition for Conformal Projection

To enable m to be independent of A in ( 6.12 ), we must satisfy the conditions that:

F ﾼ 0, E ﾼ G.

Inserting ( 6.8 ) gives:

=

∂ x

q ∂

x
∂ l þ ∂ q ∂

y y

l ﾼ

0

∂ ∂ ∂

0 1 0 1 0 1 0 1

: ð 6 : 13 Þ

2 2 2 2

;

∂ x ∂ y ∂ x ∂ y

∂

@ A @ A @ A @ A

þ ﾼ þ

∂ q ∂ q ∂ l l

From the first equation in ( 6.13 ), we have: