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Thus we are led to the series representation of the boundary condition, namely
N→∞
N→∞
d
r
q
r
=
α
0
+
α
1
q
+
α
2
q
2
+
α
r
q
r
,
x
{
q,p
=0
}
=
rB
0
+
rb
=
rB
0
+
r
(D.54)
r
=1
r
=3
N→∞
β
2
q
2
β
r
q
r
= 0
y
{
q,p
=0
}
=
β
0
−
β
1
q
−
−
(D.55)
r
=3
⇔
α
0
=
rB
0
,α
1
=
rd
1
,α
2
=
rd
2
, ··· ,α
r
=
rd
r
;
β
0
=
β
1
=
β
2
=
···
=
β
r
=0
.
(D.56)
This leads to the local representation of the transverse Mercator projection in terms of the incre-
mental isometric longitude/latitude
p/q
,namely
1
2
r
cos
2
B
0
tan
B
0
{q
2
− p
2
)+
···,
x
(
q,p
)=
rB
0
+
r
cos
B
0
q −
(D.57)
r
cos
2
B
0
tan
B
0
qp
y
(
q,p
)=
r
cos
B
0
p
−
−···
.
Once we
remove
the incremental isometric longitude/latitude
p/q
, respectively, in favor incremen-
tal longitude/latitude
l/b
, respectively, with respect to the
first chart
, we gain
2
r
cos
2
B
0
tan
B
0
N→∞
l
2
N→∞
1
c
r
b
r
c
r
c
s
b
r
b
s
x
(
q,p
)=
rB
0
+
r
cos
B
0
−
−
−···
,
(D.58)
r
=1
r,s
=1
N→∞
r
cos
2
B
0
tan
B
0
c
r
b
r
l
y
(
q,p
)=
r
cos
B
0
l
−
−···
.
r
=1
How does the local representation of the transverse Mercator projection reflect its representation
in closed form? We only have to apply a
Taylor series expansion
around
{B
0
,L
0
}
or around
{Q
0
,L
0
}
to arrive at (
D.58
).
End of Example.
Example D.2 (
S
r
, transverse Mercator projection).
2
As an example, let us construct the transverse Mercator projection
locally
for the sphere
r
based
on the fundamental solution of the d'Alembert-Euler equations (Cauchy-Riemann equations) in
terms of separation of variables. Let us depart from the equidistant mapping of the
L
0
meta-
equator, namely the
boundary condition
S
x
=
x
{
q
(
L
=
L
0
,B
)
,p
(
L
=
L
0
,B
)
}
=
rB, y
=
y
{
q
(
L
=
L
0
,B
)
,p
(
L
=
L
0
,B
)
}
=0
.
(D.59)
There remains the task to express the boundary conditions in the function space. Here, we depart
from (
D.60
). A Taylor series expansion of
B
(
Q
)for
Q ≥
0 (northern hemisphere),
u
:= exp
Q
,
leads to (
D.61
).
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