Coordinates - Map Projections: Cartographic Information Systems - page 155

cos A = cos Φ

cos B cos( Λ

− Ω ) . (3.76)

(ii) The third identity:
sin B = − cos Φ sin I sin( Λ − Ω )+sin Φ cos I.

(3.77)

Box 3.14 (From the meta-equatorial (oblique) frame of reference { E 1 , E 2 , E 3 |O} to the

equatorial frame of reference { E 1 , E 2 , E 3 |O} : the inverse transformation).

(i) The first identity:

X
R cos Φ ,

X = R cos Φ cos Λ ⇒ cos Λ = (3.78)

1
cos Φ (cos B cos A cos Ω

cos Λ = − cos B sin A sin Ω cos I +sin B sin Ω sin I ) .

(ii) The second identity:

Y
R cos Φ ,

Y = R cos Φ sin Λ ⇒ sin Λ = (3.79)

1

sin Λ =

cos Φ (cos B cos A sin Ω +cos B sin A cos Ω cos I − sin B cos Ω sin I ) .
(iii) The third identity:

sin Φ = Z

R ,
sin Φ =cos B sin A sin I +sin B cos I.

Z = R sin Φ ⇒

(3.80)

Box 3.15 (The backward problem of transforming spherical frames of reference. Input vari-
ables: A, B, Ω, I . Output variables: Λ, Φ ).

(i) The first and second identities:
tan Λ = cos B cos A sin Ω +cos B sin A cos Ω cos I

− sin B cos Ω sin I

cos B sin A sin Ω cos I +sin B sin Ω sin I .

(3.81)

cos B cos A cos Ω −

(ii) The third identity:
sin Φ =cos B sin A sin I +sin B cos I. (3.82)

3-33 The Transverse Frame of Reference of the Sphere: Part One

In the framework of the transverse aspect , we are aiming at establishing a special oblique frame of
reference by the inclination I =90 ◦ . We have to deal with two problems depending on the input
data.