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In-Depth Information
ln
1+
y
1
+
ln
1+
E
sin
x
1
E
sin
x
1
4
E
1
1
1+
y
1
4
E
y
−
=
−
y
1
−
−
E
sin
x
0
.
2
E
sin
x
1
− E
2
sin
2
x
+
(14.28)
End of Proof.
To this end, we review the mapping equations and the principal stretches. They are easily com-
puted as follows.
=
,
x
y
A
1
Λ
ln
1+
E
sin
Φ
1
−E
2
sin
2
Φ
1
−E
sin
Φ
+
(14.29)
A
1
(1
−
E
2
)
4
E
2
E
sin
Φ
Λ
1
=
1
−
E
2
sin
2
Φ
cos
Φ
cos
Φ
1
,Λ
2
=
.
(14.30)
E
2
sin
2
Φ
−
14-24 Summary (Cylindric Mapping Equations)
For the convenience of the reader, the central formulae that specify the mapping equations and
the principal stretches are summarized in the following Box
14.1
.
Box 14.1 (Summary).
Type 1 (equidistant on the set of parallel circles):
x
=
A
1
Λ, y
=
f
(
Φ
)
,
(14.31)
f
(
Φ
)=
A
1
E
(
π/
2
,E
)
E
π
arctan
A
2
A
1
tan
Φ, E
,
−
2
−
(14.32)
Λ
1
=
1
E
2
sin
2
Φ
cos
Φ
−
, Λ
2
=1
.
(14.33)
“Elliptic integral”.
Type 2 (normal conformal):
x
=
A
1
Λ, y
=
f
(
Φ
)
,
(14.34)
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