15-1 The equations governing conformal mapping 317
Example 15.1 (Universal Mercator Projection (UMP)).
x = A 1 L ,
y = A 1 ln
tan
π
4
+
B
2
1 − E sin B
1 + E sin B
E/2
.
(15.10)
The matrix of the metric of the ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
is represented by
G MN =
G 11 G 12
G 12 G 22
=
⎡
⎣
A
2
1 cos
2 B
1−E 2 sin 2 B
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 B) 3
⎤
⎦ .
(15.11)
The mapping equations of type UMP imply
x L = A 1 , x B = 0 , y L = 0 , y B =
A 1 (1 − E
2 )
(1 − E 2 sin
2 B) cos B
.
(15.12)
Korn–Lichtenstein equations:
x L =
G 11
G 22
y B , x B = −
G 22
G 11
y L , y L = −
G 11
G 22
x B , y B = −
G 22
G 11
x L ,
G 11
G 22
=
1 − E
2 sin
2 B
1 − E 2
cos B ⇒ y B =
A 1 (1 − E
2 )
(1 − E 2 sin
2 B) cos B
.
(15.13)
Integrability conditions:
∆ LB x =
G 11
G 22
x B
B
+
G 22
G 11
x L
L
= 0 , ∆ LB y =
G 11
G 22
y B
B
+
G 22
G 11
y L
L
= 0 , (15.14)
G 11
G 22
x B = 0 ,
G 22
G 11
x L =
A 1 (1 − E
2 )
(1 − E 2 sin
2 B) cos B
,
G 22
G 11
x L
L
= 0 ,
G 11
G 22
y B = A 1 ,
G 22
G 11
y L = 0 ,
G 11
G 22
y B
B
= 0 .
(15.15)
Orientation preserving conformeomorphism:
x L x B
y L y B
= (x L y B − x B y L ) =
A
2
1 (1 − E
2 )
(1 − E 2 sin
2 B) cos B
> 0 ,
(15.16)
due to −π/2 < B < +π/2 → cos B > 0.
End of Example.
The UMP solution of the Korn–Lichtenstein equations subject to the vectorial Laplace–Beltrami
equations as integrability conditions and the condition of orientation conservation is based upon the
constraint of the following type: map the equator equidistantly, for instance, x(B = 0) = A 1 Λ.