Coordinates - Map Projections: Cartographic Information Systems - page 151

1
cos φ (cos β cos α cos φ 0 sin λ 0 −

cos λ = cos β sin α sin λ 0

+sin β cos φ 0 cos λ 0 ) ,

(3.60)

1
cos φ (cos β cos α sin φ 0 sin λ 0 +cos β sin α cos λ 0
+sin β cos φ 0 sin λ 0 ) .
(ii) The third identity:
sin φ = − cos β cos α cos φ 0 +sin β sin φ 0 .

sin λ =

(3.61)

3-32 A Second Design of an Oblique Frame of Reference of the Sphere

The second design of an oblique frame of reference of the sphere S

2 R is taking reference to the

following design aspects. (i) We intersect the sphere S , generating the

oblique circular equator, also called meta-equator . We attach the oblique orthonormal frame of
reference { E 1 , E 2 , E 3 |O} at the origin O to the central plane P

2 R by a central plane P 2

O

such that { E 1 , E 2 |O} span the
central plane as well as E 3 , its unit normal vector. We connect the conventional equatorial frame
of reference

2
O

by means of the Kepler elements Ω and I , also called right ascension
of the ascending node Ω and inclination I . These Kepler elements constitute the Euler rotation
matrix R( Ω, I ):=R 1 ( I )R 3 ( Ω ). (ii) Finally, based upon such a connection, the coordinates of
the placement vector x have to be represented both in the conventional equatorial frame of
reference and in the oblique equatorial frame of reference. (iii) The forward equations as well as
the backward equations of transformation between them have to be derived.

{ E 1 , E 2 , E 3 |O}

Solution (the first, the second, and the third problem).

The three problems, in particular, can be solved (i) by representing the placement vector
X ( Λ, Φ, R ) in terms of the chosen spherical coordinates { Λ, Φ, R } with respect to the equato-

, (ii) by transforming to an oblique frame of reference
{ E 1 , E 2 , E 3 |O} by means of the Kepler elements (special Cardan angles) Ω and I , called
longitude Ω of the ascending node and inclination I , and (iii) by introducing oblique spheri-
cal coordinates {A, B} , called meta-longitude A and meta-latitude B . The complete program is
outlined in Boxes 3.10 - 3.15 . In particular, we transform from the original equatorial frame of ref-
erence { E 1 , E 2 , E 3 |O} to the oblique frame of reference, called meta-equatorial ,byR 1 ( I )R 3 ( Ω ).
Indeed, we perform a first rotation by the Cardan angle Ω around the 3 axis ( Z axis) and
a second rotation by the Cardan angle I around the 1 axis ( X axis) in order to generate
[ E 1 , E 2 , E 3 ] ∗ =R 1 ( I )R 3 ( Ω )[ E 1 , E 2 , E 3 ] ∗ . Such a procedure can be interpreted as follows: inter-
sect

rial frame of reference { E 1 , E 2 , E 3 |O}

2 R by a centric plane 2

O

to produce a circular meta-equator, which is oriented by Ω and I .
For geometrical details, consult Fig. 3.8 .

S P

End of Solution (the first, the second, and the third problem).