Coordinates - Map Projections: Cartographic Information Systems - page 157

sin Φ

1+cos Φ sin( Λ − Ω ) 1 − cos Φ sin( Λ − Ω ) .
(ii) Meta-eta-latitude:
sin B =

=

− cos Φ sin( Λ − Ω ) . (3.84)

(iii) Substitutions:

1
√ 1+tan 2 A , sin A =

tan A
√ 1+tan 2 A ,

cos A =

Ω )
cos 2 ( Λ − Ω )+tan 2 Φ , sin A =

cos( Λ − tan Φ

cos 2 ( Λ − Ω )+tan 2 Φ ,

cos A = (3.85)

1
cos B =

1

1

cos 2 Φ sin 2 ( Λ

− − Ω )

1

1

Ω ) 1

=

Ω ) . − cos Φ sin( Λ − − cos Φ sin( Λ −

Box 3.17 (The backward problem of transforming spherical frames of reference: the transverse
aspect. Input variables: A, B, Ω, I = π/ 2. Output variables: Λ, Φ ).

(i) Longitude:

tan B
cos A .

tan( Λ − Ω )= − (3.86)

(ii) Latitude:
sin Φ =cos B sin A. (3.87)

(iii) Substitutions:

sin B
cos Φ ,

sin( Λ − Ω )= −

Ω )=+ tan Φ

cos( Λ −

tan A ,

(3.88)

sin B
sin Φ

sin A
cos A = −

tan B
cos A .

tan( Λ − Ω )= −

3-34 The Transverse Frame of Reference of the Sphere: Part Two

The transverse case is a special case of an oblique frame of reference. Since it has gained great
interest in map projections, we devote another special section to the transverse aspect. In short,
for such a peculiar aspect, the meta-North Pole is chosen to be located in the conventional equator
of the reference sphere

r . In short, the spherical latitude φ 0 =0 ◦ of the meta-North Pole is fixed
to zero. Accordingly, we specialize the forward and backward transformation formulae according

S