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In-Depth Information
5-242 Case 2: Equatorial Plane of Reference
This situation is shown in Fig.
5.11
. With reference to Boxes
5.9
and
5.10
, we derive the general
form of the parameterized mapping
r
=
f
(
Δ
).Itmaybenoticednewlythat
Q
=
π
(
P
)isthe
point generated by an orthogonal projection of the point
P
2
∈
S
R
onto the axis of symmetry
North-Pole-South-Pole. Let us refer to the following identities.
Identity (i):
QP
=
R
cos
Φ.
Identity (ii):
O
∗
O
=
D
=
R
+
H.
(5.55)
Identity (iii):
O
∗
Q
=
O
∗
O
+
O
Q
=
D
+
R
sin
Φ.
Solving the perspective ratio for
r
, we are finally led to
r
=
f
(
Δ
). There is the special case
O
∗
= S, namely the identity of the perspective center
O
∗
and the South Pole S, a case that is
treated in all textbooks of Differential Geometry. Here, the distance
D
is identical to the radius
R
of the reference sphere
S
2
R
. Indeed, for this special case, we probe
r
=
R
tan
Δ/
2. Finally, we
compute
f
(
Δ
), a formula going into the computation of the left principal stretches.
Box 5.9 (Basics of the perspective ratio, equatorial plane of reference).
Basic ratio:
QP
=
O
∗
O
r
O
∗
Q
.
(5.56)
Explicit spherical representation of the basic ratio:
r
R
cos
Φ
=
D
R
sin
Φ
+
D
⇒
D
R
sin
Φ
+
D
R
cos
Φ
r
=
(5.57)
⇒
D
R
cos
Δ
+
D
R
sin
Δ.
r
=
O
∗
=S
,D
=
R
:
Special case
1+cos
Δ
, r
=
R
tan
π
=
R
tan
Δ
R
cos
Φ
1+sin
Φ
=
R
sin
Δ
Φ
2
r
=
4
−
2
.
(5.58)
Derivative of the function
r
=
f
(
Δ
) with respect to
colatitude (polar distance
Δ
):
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