Geography Reference
In-Depth Information
Indeed, for an ellipsoid-of-revolution, there exist also three aspects which are called normal,
oblique ,and transverse . Again, these aspects generate special ellipsoidal coordinates of the
ellipsoid-of-revolution, taking into account one Killing vector of symmetry . The oblique frame
of reference of
A 1 ,A 2
2
O
E
is based upon the centric oblique plane
P
, which intersects the ellipsoid-
2
O
A 1 ,A 2
of-revolution and passes the origin
O
. Such an oblique plane
P
intersects
E
in an elliptic
oblique equator, also called meta-equator , which is oriented by two Kepler elements
, called
the longitude Ω of the ascending node and the inclination I . The transverse frame of reference is
obtained by choosing an inclination I = π/ 2.
{
Ω, I
}
3-41 The Direct and Inverse Transformations of the Normal Frame
to the Oblique Frame
along the principal axes of the
ellipsoid-of-revolution of semi-major axis A 1 and semi-minor axis A 2 :
Let us orientate a set of orthonormal base vectors
{
E 1 , E 2 , E 3
}
2
3
| ( X 2 + Y 2 ) /A 1 + Z 2 /A 2 =1 ,A 1 >A 2 ,A 1 R
+ ,A 2 R
+
E
A 1 ,A 2 := { X R
}.
(3.100)
Against this frame of reference { E 1 , E 2 , E 3 |O} at the origin O , we introduce the oblique frame
of reference { E 1 , E 2 , E 3 |O} at the origin O built on an alternative set of orthonormal base
vectors which are related by means of a rotation:
E 1
E 2
E 3
E 1
E 2
E 3
=R 1 ( I )R 3 ( Ω )
.
(3.101)
This rotation is illustrated by Fig. 3.8 . The rotation around the 3 axis is denoted by Ω ,theright
ascension of the ascending node, while the rotation around the intermediate 1 axis is denoted by
I , the inclination. R 1 and R 3 are orthonormal matrices such that the following relation holds:
cos Ω sin Ω 0
sin Ω cos Ω 0
0
10 0
0cos I sin I
0 sin I cos I
R 1 ( I )R 3 ( Ω )=
0 1
cos Ω sin Ω 0
sin Ω cos I cos Ω cos I sin I
sin Ω sin I
,
=
(3.102)
cos Ω sin I cos I
3 × 3 .
R 1 ( I )R 3 ( Ω )
R
Accordingly, the following vector equation defines a representation of the placement vector X in
the orthonormal bases
{
E 1 , E 2 , E 3 }
and
{
E 1 , E 2 , E 3 }
, respectively:
3
3
E i X i .
E i X i = E 1 X + E 2 Y + E 3 Z = E 1 X + E 2 Y + E 3 Z =
X =
(3.103)
i =1
i =1
Note that the corresponding Cartesian coordinate transformations are dual to the following sys-
tems of coordinate transformations:
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