Geography Reference
In-Depth Information
Fig. 10.1.
Mapping the sphere to a (tangent) cylinder. Polar aspect. Line-of-contact: equator
There are two basic postulates which govern the setup of general equations of mapping the sphere
S
2
R
. First, the coordinate
x
depends only on the
longitude
Λ
and the parallel circles
Φ
=
±Φ
0
have to be mapped equidistantly, i.e.
x
=
RΛ
cos
Φ
0
.
Second, the coordinate
y
is only a function of latitude
Φ
, i.e.
y
=
f
(
Φ
), compare with Fig.
10.2
for
the case of a tangent cylinder. In case of the tangent variant, the cylinder is wrapping the sphere
with the equator being the line-of-contact. In the second case of a secant cylinder, two parallel
circles
Φ
=
±Φ
0
are the lines-of-contact, compare with Fig.
10.3
.
2
R
of radius
R
to a tangent or secant cylinder
C
Box 10.1 (“Sphere to cylinder”: distortion analysis, polar aspect, left principal stretches).
Parameterized mapping:
x
=
RΛ
cos
Φ
0
, y
=
f
(
Φ
)
.
(10.1)
Left Jacobi matrix:
J
l
:=
D
Λx
D
Φx
=
R
cos
Φ
0
0
.
(10.2)
D
Λ
yD
Φy
0
f
(
Φ
)
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