Geography Reference
In-Depth Information
tan
x
√
1+tan
2
x
,
sin
x
=
(1.341)
sin
l
versus sin
r
=
±
Λ
1
−
Λ
2
Λ
1
+
Λ
2
λ
1
−
λ
2
=
±
λ
1
+
λ
2
.
Maximal angular distortion:
Ω
l
:=
+
l
=2
+
l
versus
Ω
r
:=
+
r
=2
+
l
−
r
−
r
,
Ω
l
=2arcsin
versus
Ω
r
=arcsin
Λ
1
Λ
2
Λ
1
+
Λ
2
−
λ
1
λ
2
λ
1
+
λ
2
−
.
(1.342)
1-15 Exercise: The Armadillo Double Projection
Exercise: the Armadillo double projection. First: sphere to torus. Second: torus to plane. The
oblique orthogonal projection.
An excellent example of a mapping from a left two-dimensional Riemann manifold to a right two-
dimensional Riemann manifold where we have to use all the power of the previous paragraphs is
the
Armadillo map modified by Raisz
, which is illustrated in Fig.
1.32
. First, points of the sphere
S
2
2
2
R
of radius
R
are mapped onto a specific torus
T
a,b
. Second, subject to
a
=
b
=
R
,
T
a,b
is
2
O
mapped as an
oblique orthogonal projection
onto a central plane
P
. Such a double projection
2
R
2
is analytically presented in Box
1.58
. The first mapping, namely
S
→
T
a,b
,isfixedbythe
postulate
, which cuts the spherical longitude
Λ
half to be gauged to the
toroidal longitude
λ
. In contrast, spherical latitude
Φ
is set identical to the toroidal latitude
φ
.
For generating the second mapping, namely
{
λ
=
Λ/
2
,φ
=
Φ
}
2
a,b
2
O
T
→
P
, subject to
a
=
b
=
R
, we rotate around
3
to
X
,Y
,Z
}∈
R
3
. In consequence, we experience an
the 2 axis by
−
β
from
{
X, Y, Z
}∈
R
{
Z
plane such that
x
=
Y
and
y
=
Z
. In this way, we have succeeded in parameterizing the double projection
S
2
a,b
onto the
Y
−
orthogonal projection of any point of the specific torus
T
2
R
2
a,b
2
O
→
T
→
P
by
{
x
(
Λ, Φ
)
,y
(
Λ, Φ
)
}
. However, we pose the following problems. (i) Determine
from the direct mapping equations
x
(
Λ, Φ
)and
y
(
Λ, Φ
)
subject to the matrix G
l
of the metric, the right matrix G
r
of the metric, the left Jacob matrix J
l
,
and the left Cauchy-Green matrix C
l
viewed in Box
1.59
. (ii) Prove that the Armadillo double
projection is not equiareal. (iii) Prove that the images of the parallel circles of the sphere are
ellipses. Determine their semi-major and semi-minor axes as well as the location of the center.
(iv) Prove that the images of the meridians of the sphere are conic sections.
the left principal stretches
{
Λ
1
,Λ
2
}
Solution (all problems).
Here are some ideas to solve the hard problems. For the second problem, we advise you to
prove the inequality det[C
l
]
=det[G
l
]. To solve the third problem, choose
Φ
= constant and
eliminate
Λ
from the direct equatioris of the mapping, for instance, sin
Λ/
2=
x/
[
R
(1 + cos
Φ
)]
Search WWH ::
Custom Search