Geography Reference
In-Depth Information
X
=cos
U
Y
=sin
U
Z
=
V
2
(23.52)
Figure
23.14
is a computer graphic illustration of the
parabolic mirror
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
Vcos
U
Vsin
U
V
2
⎣
⎦
=
R
3
(
−γ
)
R
2
(
−β
)
R
1
(
−α
)
⎣
⎦
=
R
3
(
−γ
)
R
2
(
−β
)
R
1
(
−α
)
⎣
⎦
(23.53)
M
2
:=
P
2
,
α
=
β
=
γ
=20
◦
Fig. 23.14.
Oblique orthogonal projection of the rotational symmetric paraboloid
The oblique orthogonal projection, in consequence, is defined by Box
23.16
where (
x, y
) are the
Cartesian coordinates of the
oblique plane
which cover
R
2
.
Box 23.16 (Oblique orthogonal projection of the paraboloid
P
2
).
x
=
Y
y
=
Z
(23.54)
Next we aim at a
conformal mapping,
an
equiareal mapping
and an
equidistant mapping
of the
paraboloid
2
with boundary
onto a cylinder
2
parallel to the 3-axis (normal placement) and
P
C
2
onto a cone
0
parallel to the 3-axis (normal placement). Consult Fig.
23.15
for an illustration of
the projection geometry.
C
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