“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 219

Lemma 5.6 (Special perspective mapping of the sphere: the Lagrange projection).

The Lagrange projection of the sphere S

2 R to the equatorial plane is parameterized by

2 cos Λ = R tan π

cos Λ,

x = R tan Δ

Φ
2

4 −

(5.98)

2 sin Λ = R tan π

sin Λ,

y = R tan Δ

Φ
2 4 −

subject to the left Cauchy-Green eigenspace
left CG eigenspace = E Λ

.

1
2cos 2 2

1
2cos 2 2

, E Φ (5.99)

The Lagrange projection is conformal.

End of Lemma.

Note that the northern hemisphere is conformally mapped from the southern projective center
S=

O ∗ , while the southern hemisphere is conformally mapped from the northern projective center
N= O ∗ , namely generating northern and southern points within a circle of radius R . The union

of these two charts generates a minimal atlas of conformal type .

Question: “What makes the Lagrange projection partic-
ularly useful when compared with the Universal Stereo-
graphic Projection (UPS)?” Answer: “It is the different
factor of conformality Λ 1 = Λ 2 : the left principal stretches
of the Lagrange projection are half of the left principal
stretches of the UPS: Λ 1 (Lagrange) = Λ 2 (Lagrange) =
1

2 Λ 1 (UPS) = 2 Λ 2 (UPS).”

Box 5.17 (Lagrange projection).

Parameterized special perspective mapping (polar coordinates):
α = Λ,

(5.100)

2 = R tan π

.

r = R tan Δ

Φ
2 4 −

Sine lemma (triangle O p S) :