362 16 “Ellipsoid-of-revolution to cylinder”: oblique aspect
Conformal coordinates x and y (isometric coordinates, isothermal coordinates) are constructed
from the “surface normal” ellipsoidal coordinates L and B as solutions of the Korn–Lichtenstein
equations (conformal change from one chart to another chart, c:cha-cha-cha)
x L
x B
=
1
G 11 G 22 − G 2
12
−G 12 G 11
−G 22 G 12
y L
y B
,
(16.6)
subject to the integrability conditions x LB = x BL and y LB = y BL or
LB x :=
G 11 x B − G 12 x L
G 11 G 22 − G 2
12
B
+
G 22 x L − G 12 x B
G 11 G 22 − G 2
12
L
= 0 ,
LB y :=
G 11 y B − G 12 y L
G 11 G 22 − G 2
12
B
+
G 22 y L − G 12 y B
G 11 G 22 − G 2
12
L
= 0 ,
(16.7)
and
x L y L
x B y B
> 0
(orientation preserving conformeomorphism) ,
(16.8)
{g µν } :=
G 11 G 12
G 12 G 22
∀ µ, ν ∈ {1, 2}
(metric of the first fundamental form of E
2
A 1 ,A 2
) .
(16.9)
LB x = 0 and LB y = 0, respectively, are called the vectorial Laplace–Beltrami equations. We here
note that a Jacobi map (16.6) can be made unique by a proper boundary condition, e. g. the equidistant map of a particular coordinate line. Examples are equidistant mappings of the circular equator
(Mercator projection) or of the elliptic meridian (transverse Mercator projection). Furthermore, we
here note that only few solutions of the Korn–Lichtenstein equations (16.6) subject to the integrability
condition (16.7) (vectorial Laplace–Beltrami equations) and the condition of orientation preservation
are known. We list two in the following.
Universal Mercator Projection (UMP):
x = A 1 L =: p UMP ,
y = A 1 ln
tan
π
4
+
B
2
1 − E sin B
1 + E sin B
E/2
=: q UMP .
(16.10)
Universal Polar Stereographic Projection (UPS):
x =
2A 1
√
1 − E 2
1 − E
1 + E
E/2
tan
π
4
−
B
2
1 + E sin B
1 − E sin B
E/2
cos L =: p UPS ,
y =
2A 1
√
1 − E 2
1 − E
1 + E
E/2
tan
π
4
−
B
2
1 + E sin B
1 − E sin B
E/2
sin L =: q UPS .
(16.11)