Geography Reference
In-Depth Information
23-1 Mapping the Torus with Boundary: Projective Geometry
of the pneu
T
2
A,B
=
S
1
A
×
S
1
B
,
the
product manifold
of two circles
S
1
A
The
pneu
, geometrically described as the
torus
∈
R
+
, is comparatively represented in terms of charts of type
oblique
orthogonal projection
versus
tangential/cylindrical,
namely of type conformal, equiareal and equidistant.
Computer graphical examples are given.
The two-dimensional torus
S
1
B
of radius
A
∈
R
+
and
A
and
2
1
A
1
1
1
T
A,B
=
S
×
S
B
,
the
product manifold
of two circles
S
A
and
S
B
of
+
and
A
+
, subject to the order
B<A
as being defined by
radius
A
∈
R
∈
R
(
√
X
2
+
Y
2
2
3
A
2
)+
Z
2
B
2
=0
,A
+
,B
+
,B<A
T
A,B
:=
{
X
∈
R
|
−
−
∈
R
∈
R
}
3
x
4
+
y
4
+
z
4
+(
x
2
+
y
2
)
z
2
2(
A
2
+
B
2
)(
x
2
+
y
2
)+2(
A
2
B
2
)
z
2
=0
,
=
{
X
∈
R
|
−
−
+
,B
+
,B<A
A
∈
R
∈
R
}
,
2
for instance represented in the
chart
{
toroidal longitude
U
, toroidal latitude
V
}∈{
R
|
0
<U<
2
π,
−
π/
2
<V <π/
2
}
with the
boundaries
V=
±
π/
2bymeansof
X
=(
A
+
B cos V
)
cos U
Y
=(
A
+
B cos V
)
Sin U
(23.1)
Z
=
B sin V
Figure
23.1
is a computer graphic illustration of the
pneu
by means of an
oblique orthogonal
projection
which is generated in the following way: A rotation around the 1-axis, 2-axis and
3-axis parameterized by means of
Cardan angles
{
α
∈
[0
,
2
π
]
,
β
∈
[0
,
π
]
,
γ
∈
[0
,
2
π
]
}
transforms
(
X,Y,Z
)into(
X
,Y
,Z
)namely
⎡
⎤
⎡
⎤
X
Y
Z
X
Y
Z
⎣
⎦
=
R
3
(
⎣
⎦
−
γ
)
R
2
(
−
β
)
R
1
(
−
α
)
⎡
⎤
(
A
+
B cos V
)
cos U
(
A
+
B cos V
)
sin U
B sin V
⎣
⎦
=
R
3
(
−
γ
)
R
2
(
−
β
)
R
1
(
−
α
)
(23.2)
2
Box 23.1 (Oblique orthogonal projection of the torus
T
A,B
).
x
=
Y
y
=
Z
(23.3)
The oblique orthogonal projection, in consequence, is defined by Box
23.1
where (
x
,
y
) are the
Cartesian
coordinates of the
oblique plane
which cover
2
.
Next we aim at a
conformal mapping
,an
equiareal mapping,
and an
equidistant mapping
of the
torus
T
R
2
A,B
with boundary onto a plane through the origin
O
which is parallel to the tangent plane
at
Z
=
±B or onto a cylinder
parallel to 3-axis (normal placement) which is in
vertical contact
with
T
2
A,B
. The mapping equations are generated in such a way that along the toroidal equator
V
= 0 the mapping onto the central plane and a circular cylinder is
equidistant.
Consult Fig.
23.2
for an illustration of the projection geometry.
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