Geography Reference
In-Depth Information
x ( Λ, Φ )= cos t ( Φ ) ,y ( Φ )= b sin t ( Φ ) . (G.40)
For a given ellipsoidal latitude Φ , the reduced latitude t is a solution of the transcendental equa-
tion ( G.41 ), which for relative eccentricity E = 0 coincides with the Kepler equation.
2 t + sin 2 t = π ln 1+ E sin Φ
2 E sin Φ
1 −E 2 sin 2 Φ
1 −E sin Φ +
.
(G.41)
ln 1+ E
2 E
1 −E 2
1 −E +
There are two variants for th e gauge of the axes a and b of the ellipse x 2 /a 2 ( Λ )+ y 2 /b 2 =1 ,a ( Λ )=
. in particular, b := A 1 2 ,a =( G.36 )(variantone)and a := A 1 ,b =( G.38 )(varianttwo).
The principal distortions Λ 1 and Λ 2 of the generalized Mollweide projection are given by ( G.24 )
inserting d t/ d Φ accordingto( G.25 )and t ( Φ ), solution of ( G.41 ), which are the eigenvalues of the
general eigenvalue problem ( c AB
Λ 2 G 4 B ) E BC = 0. The principal distortions are plotted along
the eigenvectors which constitute the eigenvector matrix
E BC
}
The coordinate lines Φ =const.( t = const.) called parallel circles are mapped onto straight
lines y = const. while the coordinate lines Λ = const. called meridians are mapped onto ellipses
x 2 /a 2 ( Λ )+ y 2 /b 2 =1 ,a ( Λ )= , a straight line for Λ = 0 (Greenwich meridian), in particular.
For relative eccentricity E = 0, the generalized Mollweide projection of the biaxial ellipsoid
{
A 1 ,A 2
E
.
2
variant one, coincides with the standard Mollweide projection of the sphere
S
R .
End of Theorem.
G-3 Examples
The following six examples illustrate by computer graphics the generalized Mollweide projection
of the biaxial ellipsoid (Figs. G.1 , G.2 , G.3 , G.4 , G.5 ,and G.6 ).
S 2 R
Fig. G.1. Standard Mollweide projection of the sphere
 
Search WWH ::




Custom Search