“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 213

Lemma 5.4 (Gnomonic projection of the sphere to the polar tangential plane).

2 R to the tangential plane at the North Pole is parame-

The gnomonic projection of the sphere S

terized by the two equations

x = R cot Φ cos Λ,

(5.82)

y = R cot Φ sin Λ,

subject to the left Cauchy-Green eigenspace

left CG eigenspace = E Λ

.

1
sin Φ , E Φ

1
sin 2 Φ

(5.83)

End of Lemma.

Box 5.15 (Gnomonic projection).
Parameterized central perspective mapping (polar coordinates):

α = Λ,
r = R tan Δ = R cot Φ.

(5.84)

Parameterized central perspective mapping (Cartesian coordinates):

x
y

= R cot Φ cos Λ .

(5.85)

sin Λ

Left principal stretches:

1
sin Φ ,

Λ 1 =

(5.86)

Λ 2 = 1

sin 2 Φ .
Left eigenvectors:

1

C 1 Λ 1 = E Λ

sin Φ (Easting) ,

(5.87)

1
sin 2 Φ (Northing) .

C 2 Λ 2 = E Φ