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from (
B
=0,
Q
=0)to(
B
0
,Q
0
) and into the
first order part
from (
B
0
,Q
0
)to(
B
,
Q
). There are
various concepts to compute the arc length of an ellipse, namely.
•
Transforming the latitude
B
integral into an
elliptic integral of the first kind
by means of
introducing the
reduced latitude B
∗
(
Hopfner 1938
).
•
By means of the
Landen transformation
(
Gerstl 1984
).
•
By means of introducing the
theta function
(
Hopfner 1938
).
•
Integration by memorial quadrature (
Roesch and Kern 2000
).
•
Convergent series expansion (
Grafarend 1995
).
Table 22.12
Equidistant mapping of the Reference Meridian
“
Northing
”
A
1
1
− E
2
y
=
B
0
dB
=
B
0
0
M
(
B
)
dB
+
B
B
0
M
(
B
)
dB
(22.80)
1
E
2
(sin sin
B
)
2
3
/
2
−
“transformation of the arc length of the Reference Meridian
L
0
from ellipsoidal latitude
B
to isometric latitude
Q
”
Q
=
ln
tan
π
E/
2
1
4
+
B
EsinB
+
EsinB
−
(22.81)
2
Q
=
lntan
π
4
+
B
E
2
ln
1
EsinB
1+
EsinB
−
−
(22.82)
2
Q
=
arctanh
sin
B
−
Earctanh
(
E
sin
B
)
(22.83)
E
2
sin
2
B
1
dQ
dB
=
1
cos
B
1
−
E
2
dB
dQ
=cos
B
1
−
E
2
sin
2
B
↔
(22.84)
−
E
2
1
−
y
=
Q
Q
0
dQ
=
Q
Q
0
A
1
cos
lam
−
1
(
Q
)
A
1
cos
B
(
Q
)
1
1
dQ
(22.85)
E
2
sin
2
B
(
Q
)
E
2
sin
2
lam
−
1
(
Q
)
−
−
Here we follow the convergent series expansion for computing the
zero order integral
according
to Table
22.13
. For the evaluation of the arc length integral
B
=0
→
B
=
B
0
we expand the
x
)
−
3
/
2
for
x
:=
E
2
sin
2
B<
1in
Taylor series
and produce termwise integration of
the
uniformly convergent series.
The result in terms of powers 1
E
2
,
E
4
,etc.
and sin
2
B
,
sin
4
B
,
etc. is given by Eq.(
22.66
). In contrast, for the first order integral
B
0
→
function (1
−
Q
there
exists a more convenient polynomial representation (powers
q
,
q
2
,etc.
). For such an approach we
implement the transformation
B
B
or
Q
0
→
Q
,Eqs.(
22.81
)-(
22.83
), namely the
Lambert function lamB
and its inverse
lam
−
1
Q
. Indeed we compute the first order derivatives
dQ/dB
or
dB/dQ
in order
toendupwiththe“
Q
-arc length” Eq.(
22.85
). Such a representation gives the input for the series
expansion of type Eqs.(
22.87
)-(
22.90
)ofTable
22.14
and Eqs.(
22.91
)-(
22.96
)ofTable
22.15
,
namely for the computation of derivatives
d
h
y/dQ
h
of order h based upon the recurrence relation
Eqs.(
22.89
)and(
22.90
). Table
22.15
contains the
Taylor coe
cients
up to the order
N
=5.
The boundary condition for the “
Northern function
”
y
(
q,l
=0)=
y
=(
q,
0) has been solved in
terms of
Taylor polynomials
generated by the equidistant mapping of the
Reference Meridian L
0
.
In terms of the fundamental solution of the
Laplace-Beltrami equation
summarized in Table
22.11
we have succeed to determine the coecients
{γ
0
,γ
1
,γ
2
,γ
3
,γ
4
,γ
5
,...γ
N
}
in Table
22.15
and
→
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