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 1 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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