“Sphere to Tangential Plane”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 238

Left principal stretches:

Λ 1 = π/ 2

B
cos B

−

, Λ 2 =1 . (6.6)

Left eigenvectors:

B
sin( π/ 2 − B ) , C 2 Λ 2 = E B .

π/ 2 −

C 1 Λ 1 = E A (6.7)

Parameterized inverse mapping:

x 2 + y 2
R 2

tan A = y

x , B = π

2 −

,

Λ 0 )= sin A tan( Λ −

tan B ,

(6.8)

cos B cos A.
Left maximum angular distortion:
Ω l =2arcsin

sin Φ = −

=2arcsin

π
2

− B − cos B

Λ 1 − Λ 2

Λ 1 + Λ 2

. (6.9)

π
2

− B +cos B

6-22 Conformal Mapping (Transverse Stereographic Projection,
Transverse UPS)

The transverse conformal mapping of the sphere to a tangential plane is easily derived with the
knowledge of the preceding paragraph. We here conveniently rewrite the mapping equations of
the normal conformal mapping of Sect. 5-22 in terms of the (meta-)coordinates meta-longitude A
and meta-latitude B . Again, we take into account the relations ( 6.1 )and( 6.2 ) between meta-
coordinates, standard spherical coordinates ( Λ, Φ ), and the coordinates Λ 0 ∈ [0 ◦ , 360 ◦ ]and Φ 0 =

0 ◦ of the meta-North Pole. Then, the setup of the mapping equations is given by Lemma 6.1 .

Lemma 6.1 (Transverse conformal mapping of the sphere to a tangential plane at the meta-North
Pole Λ 0

[0 ◦ , 360 ◦ ] ,Φ 0 =0 ◦ ).

∈

x =2 R tan π

cos A, y =2 R tan π sin A,

B
2

B
2

4 − 4 −

(6.10)

tan A = sin( Λ

Λ 0 )
− tan Φ

−

, sin B =cos Φ cos( Λ − Λ 0 ) ,