176 5 “Sphere to tangential plane”: polar (normal) aspect
5-241 Case 1: northern tangential plane (tangential plane at maximal distance)
This situation is shown in Fig. 5.10. With reference to Boxes 5.7 and 5.8, we derive the general form
of the parameterized mapping r = f (∆). Note that 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. Let us refer to the
following identities.
Identity (i):
QP = R cos Φ .
Identity (ii):
O
∗ N = R + D = 2R + H .
Identity (iii):
O
∗ Q = O
∗ O + OQ =
= D + R sin Φ = R(1 + sin Φ) + H .
(5.45)
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 5.7 (Basics of the perspective ratio, northern tangential plane T N S
2
R ).
Basic ratio:
r
QP
=
O
∗ N
O ∗ Q
.
(5.46)
Explicit spherical representation of the basic ratio:
r
R cos Φ
=
R + D
R sin Φ + D
⇒
r =
R + D
R sin Φ + D
R cos Φ
⇒
r =
R + D
R cos ∆ + D
R sin ∆ =: f (∆) .
(5.47)
Derivative of the function r = f (∆) with respect to colatitude (polar distance ∆):
f
(∆) =
df
d∆
=
= R(R + D)
R + D cos ∆
(R cos ∆ + D) 2 =
= R(R + D)
R + D sin Φ
(R sin Φ + D) 2 .
(5.48)