Coordinates - Map Projections: Cartographic Information Systems - page 158

to Boxes 3.6 and 3.9 to such a special pole configuration. In order to understand better the
conventional choice of the transverse frame of reference, we additionally consider the following
example.

Example 3.7 (On the transverse frame of reference).

Let us choose λ 0 = 270 ◦ and φ 0 =0 ◦ , namely a placement of the meta-North Pole in the West
Pole. For such a configuration, the meta-equator (then called transverse equator ) agrees with the
Greenwich meridian of reference. e 1 0 = e S is directed to the South, e 2 0 = e E is directed to the
East. Such an oblique frame of reference has not found the support of traditional map projectors.
They prefer a transverse frame of reference

, namely a right-handed orthonormal
frame of reference that is oriented “East, North, Vertical” and relates to

{ e 1 ∗ , e 2 ∗ , e 3 ∗ |O}

{ e 1 0 , e 2 0 , e 3 0 |O} by

e 1 ∗ =+ e 2 0 = e E (“Easting”) ,
e 2 ∗ = − e 1 0 = e S (“Northing”) ,

(3.89)

e 3 ∗ = − e 3 0 = e V (“Vertical”) ,

In terms of this example, we may alternatively choose λ 0 = λ 0 − 270 ◦ =90 ◦ + λ 0 or λ 0 = 270 ◦ + λ 0 ,

and α = α S =90 ◦ + α E or α E = α S − 90 ◦ = 270 ◦ + α S , such that λ 0 =0 ◦ ,φ 0 =0 ◦ }

{ identifies the

270 ◦ . Greenwich meridian of reference. Indeed, we have shifted both λ 0 and α E for 3 π/ 2 ∼

End of Example.

λ, φ ; λ 0 ,φ 0 can then be conveniently solved,

namely with the result that is presented in Box 3.6 . First, for the choice φ 0 =0 , sin( λ − λ 0 )=
cos( λ − λ 0 ) , cos( λ − λ 0 )= − sin( λ − λ 0 ), and tan α =tan α S = − 1 / tan α E = − cot α E ,we
have derived the transverse equations of reference. Second, if we substitute these identities into
the representative formulae “from the equatorial frame of reference { e 1 , e 2 , e 3 |O} to the meta-
equatorial (“oblique”) frame of reference { e 1 0 , e 2 0 , e 3 0 |O} ” that are collected in Box 3.5 ,we
are directly led to the basic identities of transforming from the equatorial reference frame to the
transverse reference frame.

Note that the direct transformation { }→{ α E ,β }

3-35 Transformations Between Oblique Frames of Reference:
First Design, Second Design

Question: “How can the two oblique frames of reference e 0
and E , called first design and second design , respectively,
be related?” Answer: “The two oblique frames of reference
can be related to each other when we allow a third rotation
in the Kepler orbital plane by means of [ E 1 , E 2 , E 3 ] ∗ =
R 3 ( ω )[ E 1 , E 2 , E 3 ] ∗ .”

Note that ω is called longitude in the meta-equatorial plane . Such an additional rotation may come
as a surprise, but without such a longitude, the oblique frames of first and second kind cannot