16-1 The equations governing conformal mapping 361
Box 16.1 (The universal oblique Mercator projection of the sphere S
2
r . α, β : meta-longitude, meta-latitude.
L, B : longitude, latitude. Ω, i : longitude, inclination of the oblique meta-equator).
x = rα =
= r arc tan
ˆ
cos i tan(L − Ω) + sin i tan B/ cos(L − Ω)
˜
,
(16.1)
y = r ln tan
„
π
4
+
β
2
«
=
= r ar tanh(sin β) =
= r ar tanh
ˆ
cos i sin B − sin i cos B sin(L − Ω)
˜
,
(16.2)
tan
x
r
=
= cos i tan(L − Ω) + sin i tan B/ cos(L − Ω) ,
(16.3)
tanh
y
r
=
= cos i sin B − sin i cos B sin(L − Ω) .
(16.4)
16-1 The equations governing conformal mapping
The equations governing conformal mapping and their fundamental solution. Korn–Lichtenstein equations
and Laplace–Beltrami equations. Universal Mercator Projection (UMP) and Universal Polar Stereographic
Projection (UPS).
We are concerned with a conformal mapping of the biaxial ellipsoid E
2
A 1 ,A 2
(“ellipsoid-of-revolution”),
“spheroid”, semi-major axis A 1 , semi-minor axis A 2 ) embedded in a three-dimensional Euclid manifold
E
3 = {R
3 , δ ij } with standard “canonical” metric {δ ij }, the Kronecker delta of ones in the diagonal,
of zeros in the off-diagonal, namely by means of
x
1 =
A 1 cos B cos L
1 − E 2 sin
2 B
,
x
2 =
A 1 cos B sin L
1 − E 2 sin
2 B
,
x
3 =
A 1 (1 − E
2 ) sin B
1 − E 2 sin
2 B
,
(16.5)
introducing “surface normal” elipsoidal longitude L as well as “surface normal” ellipsoidal latitude
B, where E
2 := (A
2
1 − A
2
2 )/A
2
1 = 1 − A
2
2 /A
2
1 denotes the first relative eccentricity. According to the
relation L, B ∈ {0 ≤ L ≤ 2π, −π/2 < B < π/2}, we exclude from the domain {L, B} North and South
Pole. Thus, {L, B} constitutes only a first chart of E
2
A 1 ,A 2
. A minimal atlas of E
2
A 1 ,A 2
based on two
charts and which covers all points of the ellipsoid-of-revolution is given by E. Grafarend and R. Syffus
(1994), in great detail.