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A
+
B
cos
V
,Λ
2
=
f
f
Λ
1
=
(23.8d)
B
C
l
− Λ
2
G
l
=
− Λ
2
B
2
2
(
A
+
B
)
2
Λ
2
(
A
+
B
cos
V
)
2
−
0
=0
↔
(23.8e)
g
2
0
A
+
B
cos
V
,Λ
2
=
g
A
+
B
Λ
1
=
(23.8f)
B
2
:=
2
Box 23.3 (Conformal mapping of the torus
M
T
A,B
onto the central plane parallel to
2
and onto the circular cylinder
2
T
π/
2
M
C
A
+
B
).
f
f
B
A
+
B
cos
V
Λ
1
(
V
)=
Λ
2
(
V
)
∀V ↔
=
(23.9a)
(
A
+
B
)
B
A
+
B
cos
V
g
=
Λ
1
(
V
)=
Λ
2
(
V
)
∀
V
↔
(23.9b)
B
2
arctan
A
−
B
+ln
c
f
2
B
√
A
2
B
2
tan
V
ln
f
=
√
A
2
(23.9c)
2
−
−
B
2
arctan
A
]
2
B
√
A
2
−
B
B
2
tan
V
f
(
V
)=
c
f
exp
[
√
A
2
(23.9d)
2
−
−
− B
2
arctan
A
+
c
g
A
+
B
−
− B
2
tan
V
B
√
A
2
√
A
2
g
(
V
)=2
B
(23.9e)
2
f
(0) =
A
+
B
=
c
f
→
(23.9f)
f
(
V
)=(
A
+
B
)
exp
− B
2
arctan
A
2
B
−
− B
2
tan
V
B
√
A
2
√
A
2
(23.9g)
2
g
(0) = 0 =
c
g
→
(23.9h)
B
2
arctan
A
A
+
B
−
B
B
2
tan
V
√
A
2
√
A
2
g
(
V
)=2
B
(23.9i)
2
−
−
Box 23.4 (Equiareal mapping of the torus
M
2
:=
T
2
A,B
onto the central plane parallel to
T
π/
2
M
2
and onto the circular cylinder
C
A
+
B
).
ff
=
B
(
A
+
B
cos
V
)
Λ
1
Λ
2
(
V
)=1
∀
V
↔
(23.10a)
B
A
+
B
(
A
+
B
cos
V
)
Λ
1
Λ
2
(
V
)=1
∀V ↔ g
=
(23.10b)
1
2
f
2
=
ABV
+
B
2
+
c
f
(23.10c)
f
=
2
ABV
+
B
2
sin
V
+2
c
f
(23.10d)
B
2
A
+
B
sin
V
+
c
g
AB
A
+
B
V
+
g
(
V
)=
(23.10e)
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