“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 203

f ( Δ )= d f

R + D cos Δ
( R cos Δ + D ) 2 = DR

R + D sin Φ
( R sin Φ + D ) 2 .

d Δ = DR

(5.59)

Box 5.10 (General normal perspective mapping of the sphere to the equatorial plane of
reference).

Parameterized mapping (polar coordinates):

D
R sin Φ + D R cos Φ =

R + H
R (1 + sin Φ )+ H cos Φ.

α = Λ, r = (5.60)

Special case H =0:
r = R tan π

= R tan Δ

Φ
2 4 − 2 .

(5.61)

Parameterized mapping (Cartesian coordinates):

=

,

x
y

=

R + H

D

R (1+sin Φ )+ H R cos Φ cos Λ
R + H

R sin Φ + D R cos Φ cos Λ
D (5.62)

R sin Φ + D R cos Φ cos Λ R (1+sin Φ )+ H R cos Φ cos Λ

x
y

=

R sin Φ + D R cos Φ cos Λ

=

R (1 + sin Φ )+ H R cos Φ cos Λ

.

D R + H

sin Λ sin Λ

Special case H =0:

x
y

= R tan π cos Λ

sin Λ

= R tan Δ
2

cos Λ
sin Λ

.

φ
2 4 −

(5.63)

Left principal stretches:

R sin Δ , Λ 2 = f ( Δ )

f ( Δ )

Λ 1 = ; (5.64)

R

f ( Δ )
R sin Φ + D =

R + H
R (1 + sin Φ )+ H ,

Λ 1 =

(5.65)

( R sin Φ + D ) 2 =( R + H ) R (1 + sin Φ )+ H sin Φ

R + D sin Φ

Λ 2 = D .

[ R (1 + sin Φ )+ H ] 2

Special case H =0:

1
1+sin Φ =

1
1+cos Δ ,

Λ 1 = Λ 2 =

Λ 1 = Λ 2 = 1
2

1
cos 2 2

= 1
2

1
cos 2 ( 4 −

2 ) .

(5.66)

φ