168 5 “Sphere to tangential plane”: polar (normal) aspect
5-22 Conformal mapping (stereographic projection, UPS)
Let us postulate a conformal mapping by means of the canonical measure of conformality, i. e. Λ 1 = Λ 2 .
Such a conformal mapping of the sphere to the tangential plane of the North Pole is illustrated by
means of Fig. 5.6 that follows after the Boxes 5.4 and 5.5.
Question.
Question 1: “How can we generate the conformal mapping equations?” Answer 1: “Following
the procedure of Boxes 5.4 and 5.5, we here depart from the general representation of Λ 1 and
Λ 2 . By means of separation of variables, the relation Λ 1 = Λ 2 leads us to df/f = d∆/ sin ∆
as the characteristic differential equations. Integration of the left side as well as of the right
side leads us to the indefinite mapping equation f (∆) = c tan ∆/2.”
Question.
Question 2: “How can we gauge the integration constant?” Answer 2: “The postulate
lim ∆→0 Λ 2 (∆) = 1 that is quoted in Box 5.5 establishes an isometry at the North Pole of the
sphere. Indeed, the limit ∆ → 0 of Λ 2 (∆) = c/(2R cos
2 ∆/2) fixes c as c = 2R. Accordingly,
the polar coordinate r = f (∆) = 2R tan ∆/2 leads to the parameterized conformal mapping
x = r(∆) cos Λ and y = r(∆) sin Λ.”
This conformal mapping is called UPS (Universal Polar Stereographic Projection) for the following
reason. Figure 5.6, which illustrates this stereographic projection, focuses on the peripheral angle
∆/2 = π/4 − Φ/2 at the South Pole. Obviously, a projection line departing from the perspective
center intersects at P ∈ S
2
R and p ∈ T N S
2
R . Compare with Lemma 5.2, which summarizes the UPS
(Universal Polar Stereographic Projection).
Lemma 5.2 (UPS, conformal mapping of the sphere to the tangential plane at the North Pole).
The conformal mapping of the sphere to the tangential plane at the North Pole, in short, UPS
(Universal Polar Stereographic Projection), is parameterized by
x = 2R tan
∆
2
cos Λ =
= 2R tan
π
4
−
Φ
2
cos Λ ,
y = 2R tan
∆
2
sin Λ =
= 2R tan
π
4
−
Φ
2
sin Λ ,
(5.24)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
cos 2
π
4 −
Φ
2
, E Φ
1
cos 2
π
4 −
Φ
2
.
(5.25)
End of Lemma.
In the case of UPS, the areal distortion increases fast with colatitude (polar distance ∆), namely
Λ 1 Λ 2 − 1 = cos
−4
π/4 − Φ/2
− 1 and Λ 1 Λ 2 − 1 → ∞ for ∆ → π, and this is the reason for the
application of UPS as outlined in the following historical aside.
5-22 Conformal mapping (stereographic projection, UPS)
Let us postulate a conformal mapping by means of the canonical measure of conformality, i. e. Λ 1 = Λ 2 .
Such a conformal mapping of the sphere to the tangential plane of the North Pole is illustrated by
means of Fig. 5.6 that follows after the Boxes 5.4 and 5.5.
Question.
Question 1: “How can we generate the conformal mapping equations?” Answer 1: “Following
the procedure of Boxes 5.4 and 5.5, we here depart from the general representation of Λ 1 and
Λ 2 . By means of separation of variables, the relation Λ 1 = Λ 2 leads us to df/f = d∆/ sin ∆
as the characteristic differential equations. Integration of the left side as well as of the right
side leads us to the indefinite mapping equation f (∆) = c tan ∆/2.”
Question.
Question 2: “How can we gauge the integration constant?” Answer 2: “The postulate
lim ∆→0 Λ 2 (∆) = 1 that is quoted in Box 5.5 establishes an isometry at the North Pole of the
sphere. Indeed, the limit ∆ → 0 of Λ 2 (∆) = c/(2R cos
2 ∆/2) fixes c as c = 2R. Accordingly,
the polar coordinate r = f (∆) = 2R tan ∆/2 leads to the parameterized conformal mapping
x = r(∆) cos Λ and y = r(∆) sin Λ.”
This conformal mapping is called UPS (Universal Polar Stereographic Projection) for the following
reason. Figure 5.6, which illustrates this stereographic projection, focuses on the peripheral angle
∆/2 = π/4 − Φ/2 at the South Pole. Obviously, a projection line departing from the perspective
center intersects at P ∈ S
2
R and p ∈ T N S
2
R . Compare with Lemma 5.2, which summarizes the UPS
(Universal Polar Stereographic Projection).
Lemma 5.2 (UPS, conformal mapping of the sphere to the tangential plane at the North Pole).
The conformal mapping of the sphere to the tangential plane at the North Pole, in short, UPS
(Universal Polar Stereographic Projection), is parameterized by
x = 2R tan
∆
2
cos Λ =
= 2R tan
π
4
−
Φ
2
cos Λ ,
y = 2R tan
∆
2
sin Λ =
= 2R tan
π
4
−
Φ
2
sin Λ ,
(5.24)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
cos 2
π
4 −
Φ
2
, E Φ
1
cos 2
π
4 −
Φ
2
.
(5.25)
End of Lemma.
In the case of UPS, the areal distortion increases fast with colatitude (polar distance ∆), namely
Λ 1 Λ 2 − 1 = cos
−4
π/4 − Φ/2
− 1 and Λ 1 Λ 2 − 1 → ∞ for ∆ → π, and this is the reason for the
application of UPS as outlined in the following historical aside.