Geography Reference
In-Depth Information
1-144 Optimal Map Projections
Optimal map projections relate to the invariant scalar measures of Cauchy-Green deformation.
More than 1,000 scientific contributions have been published on this topic. Harmonic maps, opti-
mal Universal Mercator Projections (opt UMP) as well as optimal Universal Transverse Mercator
(opt UTM) belong to this category. Let us only introduce here the
optimality conditions
as they
are summarized in B
ox
es
1.50
-
1.52
. First,
Airy
(
1861
),
Kavrajski
(
1958
) introduced local as well
as global measures of
f
:
M
r
from isometry. Since for an isometry canonically
Λ
1
=
Λ
2
=1
or
λ
1
=
λ
2
=1holds,
{Λ
1
−
1
,Λ
2
−
1
}
or
{
ln
Λ
1
,
ln
Λ
2
}
and
{λ
1
−
1
,λ
2
−
1
}
or
{
ln
λ
1
,
ln
λ
2
}
as
“errors”
l
and
r
are measures of the local departure from isometry. When integrated over the
part of the left or right surface to be mapped, we are led to the global measures of departure from
isometry, na
m
ely
I
A
and
I
AK
of type “left” and “right”. Second, we introduce local and global
measures of
f
:
M
l
→
M
r
from an areomorphism or a conformeomorphism. Since for an equiareal
mapping canonically
Λ
1
Λ
2
=1or
λ
1
λ
2
=1holds,
l
→
M
as “errors”
l
and
r
of type “areal” measure the local departure from an areomorphism. Similarly, for a conformal
mapping canonically
Λ
1
=
Λ
2
or
λ
1
=
λ
2
holds. Accordingly,
Λ
1
−
{
Λ
1
Λ
2
−
1
}
or
{
λ
1
λ
2
−
1
}
λ
2
as (errors
l
and
r
as measures of type “conformal” describe the local departure from a conformeomorphism. When
integrated over the part of the left or right surface to be mapped, we are led to global measures of
departure from areomorphism or conformeomorphism, namely
I
areal
and
I
conf
of type “left” and
“right”. Examples are given in the following chapters.
Λ
2
or
λ
1
−
2
l
r
from isometry).
Box 1.50 (Local measures for departure of the mapping
M
→
M
(i)
Airy
(
1861
):
l
A
:
1
1)
2
] sus
1
1)
2
+(
Λ
1
1)
2
+(
λ
2
1)
2
]=:
rA
.
2
[(
Λ
1
−
−
2
[(
λ
1
−
−
(1.312)
(ii)
Kavrajski
(
1958
):
l
AK
:=
1
2
[(ln
Λ
1
)
2
+(ln
Λ
2
)
2
] sus
1
2
[(ln
λ
1
)
2
+(ln
λ
2
)
2
]=:
rAK
.
(1.313)
2
l
2
Box 1.51 (Local measures for departure of the mapping
M
→
M
r
from equiareal and
conformal).
(i) Departure from an equiareal mapping:
l
areal
:= (
Λ
1
Λ
2
−
1)
2
1)
2
=:
rareal
.
versus (
λ
1
λ
2
−
(1.314)
(ii) Departure from a conformal mapping:
l
conf
:= (
Λ
1
−
Λ
2
)
2
λ
2
)
2
=:
rconf
.
versus (
λ
1
−
(1.315)
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