“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 195

Lemma 5.3 (Equiareal azimuthal projection of the sphere to the tangential plane at the North
Pole).

The equiareal mapping of the sphere to the tangential plane at the North Pole is parameterized
by the two equations

x =
=2 R sin Δ

2 cos Λ =

=2 R sin π

cos Λ,

Φ
2

4 −

(5.35)

y =
=2 R sin Δ

2 sin Λ =

=2 R sin π

sin Λ,

Φ
2 4 −

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

.

2 , E Φ cos π

1 Φ

2 cos 4 −

4 −

(5.36)

Φ

End of Lemma.

From the sketch that is shown in Fig. 5.8 , we gain some geometric understanding of how to
construct the normal equiareal mapping by a “pair of dividers and a ruler”. The radial coordinate
r =N p coincides with the segment N P =2 R sin Δ/ 2, the peripheral point P within the vertical
section constitutes a rectangular triangle NS P subject to SN = 2 R .

Box 5.6 (Equiareal mapping of the sphere to the tangential plane at the North Pole).

Postulate of a areomorphism:
Λ 1 Λ 2 =1 ,

f ( Δ )
R 2 sin Δ

d f ( Δ )
d Δ

f ( Δ )d f ( Δ )= R 2 sin Δ d Δ.

=1 ⇒ (5.37)

Integration of the characteristic differential equation of an
equiareal mapping

2
R

2

S → T N S

R subject to an initial condition:

f 2 = −R 2 cos Δ + c
r (0) = f (0) = 0

⇒ 0= −R 2 + c ⇒ c = R 2 ,

(5.38)

f 2 =2 R 2 (1

cos Δ )=4 R 2 sin 2 2

r = f ( Δ )=2 R sin Δ

−

⇒

2 .

2sin 2 2

cos x =1 −