180 5 “Sphere to tangential plane”: polar (normal) aspect
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 ∈ S
2
R onto the axis of symmetry North-Pole–South-Pole. Note
that the southern projection plane is at distance D from the origin O or, alternatively, at spherical
height H from O
∗ . Collected in Box 5.11, we present to you the basic identities
Identity (i):
QP = R |cos Φ| = R |sin ∆| .
Identity (ii):
O
∗ S = D − R = H .
Identity (iii):
O
∗ Q = O
∗ O + OQ =
= D − R |sin Φ| = R(1 − |sin Φ|) + H .
(5.67)
Solving the perspective ratio for r, we are finally led to r = f (∆). Such a representation of the radial
function f (∆) is supplemented by the computation of f
(∆), 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:
r
QP
=
O
∗ S
O ∗ Q
.
(5.68)
Explicit spherical representation of the basic ratio:
r
R cos Φ
=
H
R + H − R |sin Φ|
=
D − R
D − R |sin Φ|
⇒
r =
HR cos Φ
H + R(1 − |sin Φ|)
=
D − R
D − R |sin Φ|
R cos Φ
⇒
r =
D − R
D − R |cos ∆|
R |sin ∆| =: f (∆) .
(5.69)
Derivative of the function r = f (∆) with respect to colatitude (polar distance ∆):
f
(∆) =
df
d∆
=
= R(D − R)
D |cos ∆| − R
(D − R |cos ∆|) 2 =
= R(D − R)
D |sin Φ| − R
(D − R |sin Φ|) 2 .
(5.70)