Geography Reference
In-Depth Information
As an alternative, we could base a second design of the transverse Mercator projection modulo
an unknown dilatation factor
ρ
on a
minimal total areal distortion
over a meridian strip [
l
W
,l
E
]
×
[
B
S
,B
N
] between a longitudinal extension
L
W
and
L
E
and a latitudinal extension
B
S
and
B
N
,in
particular, the distortion measure (
15.106
) with respect to principal distortions
Λ
1
and
Λ
2
,and
the spheroidal surface element. But surprisingly, it does not differ from the one of the previous
corollary, the optimal Airy design, see Corollary
15.8
, the proof of which we want to leave to the
reader.
I
l
(areal) :=
(15.106)
d
S
(
Λ
1
Λ
2
−
:=
1
S
1)
.
Corollary 15.8 (The dilatation factor for an optimal universal transverse Mercator projection,
the zero total distortion).
[
B
S
,B
N
] of type universal trans-
verse Mercator projection, the postulate of
minimal total distance distortion
(Airy optimum)
and the postulate of
minimal total areal distortion
lead to the same unknown dilatation factor
ρ
(
For a conformal map of the half-symmetric strip [
−
l
E
,l
E
]
×
→
(
15.98
)-(
15.100
)). The total areal distortion amounts to zero!
End of Corollary.
For this chapter and the other chapters, please consult the following publications.
Airy
(
1861
),
Amalvict and Livieratos
(
1988
),
Beltrami
(
1869
),
Blaschke and Leichtweiß
(
1973
),
Bretter-
bauer
(
1980
),
Do Carmo et al.
(
1985
),
Cauchy
(
1823
,
1828
),
Clarke and Helmert
(
1911
),
Cole
(
1943
),
Cox et al.
(
1996
),
Dermanis and Livieratos
(
1983a
,
1993
),
Dermanis et al.
(
1984
),
Engels and Grafarend
(
1995
),
Euler
(
1755
,
1770
),
Finzi
(
1922
),
Gauss
(
1813
,
1816-1827
-
1827
,
1822
,
1844
),
Glasmacher and Krack
(
1984
),
Goenner et al.
(
1994
),
Grafarend
(
1995
),
Grafarend
and Syffus
(
1995
,
1998c
),
Green
(
1841
),
Hedrick and Ingold
(
1925a
,
b
),
Heitz
(
1988
),
Hotine
(
1946
,
1947a
,
b
,
c
,
d
),
Jacobi
(
1839
),
Kaltsikis
(
1980
),
Kavrajski
(
1958
),
Klingenberg
(
1982
),
Koenig
and Weise
(
1951a
),
Korn
(
1914
),
Krueger
(
1903
,
1912
,
1914
,
1922
),
Kulkarni
(
1969
,
1972
),
Kulkarni and Pinkall
(eds.,
1988
),
Laborde Chef d'escadron
(
1928
),
Lafontaine
(
1988a
,
b
)
de
Lagrange
(
1781
),
Lee
(
1944
,
1976
),
Lichtenstein
(
1911
,
1916
),
Lilienthal
(
1902
),
Liouville
(
1850
),
Livieratos
(
1987
),
van Loan
(
1976
),
Maling
(
1959/1960
,
1973
),
Maurer
(
1935
),
Markusche-
witsch
(
1955
),
Miller
(
1941
),
Misner
(
1978
),
Mitra and Rao
(
1968
),
De Moor and Zha
(
1991
),
Mueller
(
1991
),
Ricci
(
1918
),
Richardus and Adler
(
1972a
,
b
),
Riemann
(
1851
),
Rosenmund
(
1903
),
Schering
(
1857
),
Schmehl
(
1927
),
Schouten
(
1921
),
Snyder
(
1979a
,
b
,
c
,
1982
),
Spallek
(
1980
),
Stein
and Weiss
(
1968
),
Ting
(
1985
),
Tissot
(
1881
),
Uhlig
(
1976
,
1979
),
Weber
(
1867
),
Wray
(
1974
),
Yano
(
1970
),
Yanushaushas
(
1982
),
Zadro and Carminelli
(
1966
),
Zha
(
1991
)and
Zund
(
1987
).
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