“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 196

Parameterized equiareal mapping:

x
y

=2 R sin Δ
2

cos Λ
sin Λ

=2 R sin π cos Λ

sin Λ

.

Φ
2

4 −

(5.39)

Left principal stretches:
f ( Δ )= R cos Δ

2 ,

2 =cos π

;

1
cos 2

1
cos( 4 −

2 ) , Λ 2 =cos Δ

Φ
2

Λ 1 = =

4 −

(5.40)

Φ

π
2 : lim

special value (isometry): Φ →

Φ→π/ 2 Λ 1 = lim Φ→π/ 2 Λ 2 =1 .

(5.41)

Left eigenvectors:

1

cos 4 −

2

C 1 Λ 1 = E Λ (“Easting”),

Φ

C 2 Λ 2 = E Φ cos π

Φ
2

4 −

(“Northing”). (5.42)

Left maximal angular distortion:
Ω l =2arcsin =2arcsin 1 − cos 2 2

1+cos 2 2

Λ 1 − Λ 2

Λ 1 + Λ 2

. (5.43)

Parameterized inverse mapping:
tan Λ = y

x ,

2 R x 2 + y 2 .

sin Δ

1

2 =

(5.44)

The geometric construction to be considered here may have
motivated Lambert ( 1772 ) to invent such an equiareal map-
ping of the sphere. Due to the postulate of an equiareal
mapping, the equiareal azimuthal projection of the sphere
is very popular in Geostatistics.

In order to complete the considerations, we present to you Fig. 5.9 , which shows a sample of the
polar equiareal projection of the sphere.