14-3 General cylindric mappings (equidistant, rotational-symmetric figure) 309
14-33 Special normal equiareal cylindric mapping (equidistant + conformal: equator)
H ere, let us depart from the the postulate of an equiareal mapping. S
tep by step, this leads to the
following result.
Λ 1 Λ 2 = 1 ⇒
F (0)
F (Φ)
f
(Φ)
F 2 (Φ) + G 2 (Φ)
= 1 ⇔ df =
F (Φ)
F (0)
F 2 (Φ) + G 2 (Φ) dΦ ,
f (0) = 0 ⇒ f (Φ) =
1
F (0)
Φ
0
F ( ˜
Φ)
F 2 ( ˜
Φ) + G 2 ( ˜
Φ) d ˜
Φ + const. ,
f (0) = 0 ⇒ const. = 0 .
(14.54)
The following formulae define the general mapping equations and the left principal stretches for an
equiareal cylindric mapping.
x
y
=
⎡
⎣
F (0)Λ
1
F (0)
Φ
0
F ( ˜
Φ)
F 2 ( ˜
Φ) + G 2 ( ˜
Φ)d ˜
Φ
⎤
⎦ , Λ 1 =
F (0)
F (Φ)
, Λ 2 =
F (Φ)
F (0)
=
1
Λ 1
. (14.55)
14-34 An example (mapping the torus)
The torus is the product manifold S
1
A × S
1
B , especially for the parameter range A > B, 0 < U < 2π,
and 0 < V < 2π.
⎡
⎢
⎢
⎣
X
Y
Z
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
(A + B cos V ) cos U
(A + B cos V ) sin U
B sin V
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
(A + B cos Φ) cos Λ
(A + B cos Φ) sin Λ
B sin Φ
⎤
⎥
⎥
⎦ .
(14.56)
The torus is the special surface which is generated by rotating a circle of radius B relative to a circle
of radius A > B around the center of the circle.
F (Φ) = A + B cos Φ , G(Φ) = B sin Φ ,
U = Λ = arctan
Y
X , V = Φ = arctan
Z
√
X 2 +Y 2 −A
.
(14.57 )
L et us summariz e in B
ox 14.3 the (left) tangent vector
s and the (left) coordinates of the metric tensor
of the torus.
Box 14.3 (A special rotational figure: the torus).
Left tangent vectors:
G Λ :=
∂X
∂Λ
= −E 1 (A + B cos Φ) sin Λ + E 2 (A + B cos Φ) cos Λ ,
G Φ :=
∂X
∂Φ
= −E 1 B sin Φ cos Λ − E 2 B sin Φ sin Λ + E 3 B cos Φ ,
(14.58)
F
(Φ) = −B sin Φ , G
(Φ) = B cos Φ .
(14.59)
Coordinates of the metric tensor:
G l =
»
(A + B cos Φ)
2
0
0
B
2
–
.
(14.60)