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f
(
Φ
)=
A
1
ln
tan
π
E/
2
,
1
4
+
Φ
E
sin
Φ
1+
E
sin
Φ
−
(14.35)
2
Λ
1
=
Λ
2
=
1
E
2
sin
2
Φ
cos
Φ
−
.
(14.36)
Type 3 (normal equiareal):
x
=
A
1
Λ, y
=
f
(
Φ
)
,
(14.37)
ln
1+
E
sin
Φ
1
− E
sin
Φ
+
,
E
2
)
f
(
Φ
)=
A
1
(1
−
2
E
sin
Φ
(14.38)
4
E
E
2
sin
2
Φ
1
−
Λ
1
=
1
E
2
sin
2
Φ
cos
Φ
−
cos
Φ
, Λ
2
=
1
.
(14.39)
E
2
sin
2
Φ
−
14-3 General Cylindric Mappings (Equidistant, Rotational-Symmetric
Figure)
General mapping equations and distortion measures of cylindric mappings of type
equidistant
mappings
in case of a rotationally symmetric figure.
Let us here review the structure of the general mapping equations of a rotationally symmetric
figure mapped onto a cylinder: in Box
14.2
, we collect the parameterization of a rotationally
symmetric figure, the left coordinates of the metric tensor, the right coordinates of the metric
tensor, the left Cauchy-Green matrix, and the left principal stretches. Following this, we present
special
cylindric mappings
of a
rotationally symmetric figure
which are equidistant on the equator.
In addition, let us assume that the image coordinate
y
depends only on the
latitude Φ
, while
the image coordinate
x
depends on the
longitude Λ
under the constraint that the equator is
mapped
equidistantly
.
x
(
Λ
)and
y
(
Φ
) are the result. Finally, special cylindric mappings onto the
rotationally symmetric figure are presented. As an example, we present the
torus
.
Box 14.2 (Rotationally symmetric figure mapped onto a cylinder).
Parameterization of a rotationally symmetric figure:
{
Λ, Φ
}→{
X,Y,Z
}
,
(14.40)
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