Geography Reference
In-Depth Information
As a basis for a discussion of the Airy optimal generalized Mercator projection (UM), we refer
to Table I.1 and Fig. I.1 where the Airy optimal dilatation factor ρ ( Φ N ) as a function of the
strip width 2 Φ N with respect to WGS 84 has been computed or plotted, respectively. Finally, we
present two examples for the optimal design of the generalized Mercator projection which can be
compared to those of Grafarend ( 1995 ) for the optimal transverse Mercator projection.
Example I.1 ([ Φ S = −Φ N N ]=[ 15 , +1 . 5 ]).
For the Airy optimal generalized UM, we have chosen a strip width of 3 between Φ S = 1 . 5
southern latitude and Φ N =1 . 5 northern latitude. Once we refer to the WGS 84, the Airy optimal
dilatation factor amounts to
ρ =0 . 999 887 .
(I.29)
End of Example.
Example I.2 ([ Φ S = −Φ N N ]=[ 3 , +3 ]).
For the second example, we have chosen a strip width of 6 between Φ S = 3 southern latitude
and Φ N =3 northern latitude. Once we refer to WGS 84, the Airy optimal dilatation factor
amounts to
ρ =0 . 999 546 .
(I.30)
End of Example.
Tab l e I . 1
Airy optimal dilatation factor ρ for a symmetric strip [ Λ W E ]
×
[ Φ S =
Φ N N ], Λ W = Λ 0
ΔΛ ,
Λ E = Λ 0 + ΔΛ , generalized UM, WGS 84, Λ 1 =6 , 378 , 137 m, E 2 =0 . 00669437999013
Φ N =1 . 5
Φ N =3
Φ N =6
Φ N =10
ρ =0 . 999887
ρ =0 . 999546
ρ =0 . 998183
ρ =0 . 994943
I-2 The Optimal Polycylindric Projection of Conformal Type (UPC)
Here, we present two definitions which relate to the generalized polycylindric projection of con-
formal type and the Airy optimal generalized polycylindric projection of conformal type (UPC).
Three lemmas describe in detail the optimal UPC for the ellipsoid-of-revolution, which is finally
illustrated by four tables and five figures, including a detailed example for the geographic region
of Indonesia. All optimal map projections refer to WGS 84.
Definition I.8 (Generalized polycylindric projection, mapping equations).
The conformal mapping of the ellipsoid-of-revolution ( I.32 ) with semi-major axis A 1 , semi-minor
axis A 2 , and relative eccentricity squared E 2 := ( A 1 −A 2 ) /A 1 onto the developed circular cylinder
C
2 R of radius ( I.31 ) is called a generalized polycylindric projection if the parallel circle-of-reference
Φ 0 is mapped equidistantly except for a dilatation factor ρ 0 such that the mapping equations ( I.33 )
hold.
 
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