Geography Reference
In-Depth Information
2
R
to the
equatorial plane of reference, specifically of the parameterized mapping in both polar coordinates
{α,r}
and in Cartesian coordinates
{x, y}
, completed by the computation of the left principal
stretches
{Λ
1
,Λ
2
}
. For the special case
O
∗
= S or, equivalently,
D
=
R
or
H
=0,weprove
conformality
Λ
1
=
Λ
2
. For such a configuration of the southern perspective center,
Φ
=
Box
5.10
is a summary of the general normal perspective mapping of the sphere
S
−
π/
2is
singular:
Λ
1
(
−
π/
2) =
Λ
2
(
π/
2)
→∞
.
5-243 Case 3: Southern Tangential Plane (Tangential Plane at Minimal Distance)
This situation is shown in Fig.
5.12
. According to Fig.
5.12
.
Q
=
π
(
P
) is the point generated by
an orthogonal projection of the point
P
2
R
onto the axis of symmetry North-Pole-South-Pole.
Note that the southern projection plane is at distance
D
from the origin
∈
S
O
or, alternatively, at
O
∗
. Collected in Box
5.11
, we present to you the basic identities
spherical height
H
from
Identity (i):
QP
=
R
|
cos
Φ
|
=
R
|
sin
Δ
|
.
Identity (ii):
O
∗
S=
D − R
=
H.
(5.67)
Identity (iii):
O
∗
Q
=
O
∗
O
+
OQ
=
=
D
−
R
|
sin
Φ
|
=
R
(1
−|
sin
Φ
|
)+
H.
Solving the perspective ratio for
r
, we are finally led to
r
=
f
(
Δ
). Such a representation of the
radial function
f
(
Δ
) is supplemented by the computation of
f
(
Δ
), a formula needed for the
analysis of the left principal stretches.
Box 5.11 (Basics of the perspective ratio, tangential plane at minimal distance to
O
∗
).
Basic ratio:
QP
=
O
∗
S
r
O
∗
Q
.
(5.68)
Explicit spherical representation of the basic ratio:
r
R
cos
Φ
=
H
D − R
=
R
+
H
−
R
|
sin
Φ
|
D
−
R
|
sin
Φ
|
⇒
HR
cos
Φ
H
+
R
(1
D − R
r
=
R
cos
Φ
)
=
(5.69)
−|
sin
Φ
|
D
−
R
|
sin
Φ
|
⇒
D
−
R
r
=
R
|
sin
Δ
|
=:
f
(
Δ
)
.
D
−
R
|
cos
Δ
|
Derivative of the function
r
=
f
(
Δ
) with respect to
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