“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 205

colatitude (polar distance Δ ):
f ( Δ )= d f

d Δ =

D | cos Δ |− R

= R ( D − R )

) 2 =

(5.70)

( D − R | cos Δ |

R
( D − R| sin Φ| ) 2 .

D | sin Φ |−

= R ( D − R )

By means of Box 5.12 , we have collected the parameter equations which characterize the general
normal perspective mapping of the sphere

2 R to the southern tangential plane, specifically in terms

S

of polar coordinates , completed by the computation of

the left principal stretches {Λ 1 ,Λ 2 } .Atthepointofsymmetry,namely Δ = π or Φ = −π/ 2, we
prove the isometry Λ 1 = Λ 2 =1.

{ α,r } and of Cartesian coordinates { x, y }

Box 5.12 (General normal perspective mapping of the sphere to the tangential plane at
minimal distance).

Parameterized mapping (polar coordinates):
α = Λ,

R cos Φ H

H + R (1

r =( D − R ) =

) R cos Φ.

(5.71)

D − R | sin Φ | −| sin Φ |

Parameterized mapping (Cartesian coordinates):
x
y

= ( D

,

R ) R cos Φ cos Λ

−

D−R| sin Φ|

R ) R cos Φ sin Λ
D−R| sin Φ|

( D −

(5.72)

x
y

=

H + R (1 −| sin Φ| ) R cos Φ cos Λ

.

H

sin Λ

Left principal stretches:

f ( Δ )
R sin Δ ,

Λ 1 =

(5.73)

Λ 2 = f ( Δ )
R

;

D − R H

H + R (1

Λ 1 = =

) ,

D − R | cos Δ | −| sin Φ |

(5.74)