14-3 General cylindric mappings (equidistant, rotational-symmetric figure) 311
The special case normal cylindric mapping, equidistant on the equator and the set of parallel
circles, the special case normal conformal cylindric mapping, equidistant on the equator, and the
special normal equiareal cylindric mapping, equidistant on the equator are summarized in Box 14.4.
Box 14.4 (Summary).
Case 1
(normal cylindric mapping, equidistant on the equator and the set of parallel circles):
f (Φ) =
Z Φ
0
BdΦ
= BΦ ,
»
x
y
–
=
»
(A + B)Λ
BΦ
–
.
(14.66)
Case 2
(normal conformal cylindric mapping, equidistant on the equator):
f (Φ) = (A + B)
Z Φ
0
B
A + B cos Φ dΦ
=
= (A + B)
Z Φ
0
dΦ
AB −1 + cos Φ =
=
2B(A + B)
√
A 2 − B 2
arctan
tan
Φ
2
p
A 2 − B 2
A + B
.
(14.67)
Mapping equations and principal stretches:
»
x
y
–
=
2
6
4
(A + B)Λ
2B(A+B)
√
A 2 −B 2
arctan
tan
Φ
2
p
A 2 − B 2
A+B
3
7
5 , Λ 1 = Λ 2 =
F (0)
F (Φ)
=
A + B
A + B cos Φ
. (14.68)
Case 3
(normal equiareal cylindric mapping, equidistant on the equator):
f (Φ) =
B
A + B
Z Φ
0
`
A + B cos Φ
´
dΦ
=
=
AB
A + B
Z Φ
0
dΦ
+
B
2
A + B
Z Φ
0
dΦ
cos Φ
=
AB
A + B
Φ +
B
2
A + B
sin Φ .
(14.69)
Mapping equations and principal stretches:
»
x
y
–
=
»
(A + B)Λ
AB
A+B
Φ +
B
2
A+B
sin Φ
–
, Λ 1 =
F (0)
F (Φ)
=
A + B
A + B cos Φ
, Λ 2 =
1
Λ 1
=
A + B cos Φ
A + B
. (14.70)