126 3 Coordinates
3-3 The oblique frame of reference of the sphere
The oblique frame of reference of the sphere (three Killing vectors): normal, oblique, and transverse
aspects, Killing symmetry, designs of an oblique frame of reference of the sphere.
Let us confront you here with three aspects of the sphere, which are called normal, oblique, and
transverse. These aspects form the basis of spherical coordinates of the sphere, taking into account
the three Killing vectors of symmetry, typical for a spherical surface. The first oblique frame of
reference of S
2
r is based upon the meta-North Pole, with the spherical coordinates {λ 0 , φ 0 } as design
elements. Alternatively, the second oblique frame of reference of S
2
R refers to the centric oblique plane
P
2
O , which intersects the sphere S
2
R and passes the origin O. The oblique plane P
2
O intersects S
2
R in a
so-called circular oblique equator, also called meta-equator, which is oriented by two Kepler elements
{Ω, I}, namely the longitude Ω of the ascending node and the inclination I. Finally, the third frame
of reference, which is called transverse, is defined as special oblique, namely by an inclination I = π/2.
For all three frames of reference, we present to you the forward and backward transformation formulae,
also called direct and inverse. Their derivation is technically done in a way which is suitable for other
figures of reference which have less Killing symmetry, like the ellipsoid-of-revolution.
3-31 A first design of an oblique frame of reference of the sphere
The first design of an oblique frame of reference of the sphere S
2
r is taking reference to the following
design aspects. (i) We make a choice about the three spherical coordinates {λ 0 , φ 0 , r} of the “new
North Pole” (which is also called meta-North Pole) relative to the conventional equatorial frame of
reference. We attach to the direction of the new North Pole a new equatorial frame of reference, which
is called meta-equatorial, namely {e 1 0 , e 2 0 , e 3 0 x 0 }, a set of orthonormal base vectors at the point
x(λ 0 , Φ 0 , r) =: x 0 , such that e 3 0 := x 0 / x 0 . We connect the conventional equatorial frame of
reference {e 1 , e 2 , e 3 O} to the oblique or meta-equatorial frame of reference {e 1 0 , e 2 0 , e 3 0 O} at the
origin O, namely by parallel transport of {e 1 0 , e 2 0 , e 3 0 } from x 0 to P O (in the Euclidean sense). (ii)
We represent the coordinates of the placement vector x, both in the conventional equatorial frame of
reference and in the oblique equatorial frame of reference. (iii) We finally derive the forward as well
as backward equations of transformation between them.
Solution (the first problem).
The first problem can be solved (i) by representing the placement vector x(λ 0 , φ 0 , r) in terms
of the chosen spherical coordinates {λ 0 , φ 0 , r}, (ii) by computing the triplet of partial derivatives
{D λ 0 x, D φ 0 x, D r x}, which are normalized by the Euclidean norms D λ 0 x , D φ 0 x , and D r x ,
leading to the triplet −e φ 0 := D φ 0 x/ D φ 0 x , +e λ 0 := D λ 0 x/ D λ 0 x , and +e r := D r x/ D r x ,
and these three triplet terms are called South, East, and Vertical. Finally, we relate the base vectors
{e 1 0 , e 2 0 , e 3 0 x 0 } := {e λ 0 , e φ 0 , e r x 0 } to {e 1 , e 2 , e 3 O}. This is outlined in Box 3.3. For geometrical
details, consult Figs. 3.6 and 3.7.
End of Solution (the first problem).
Solution (the second problem).
The second problem, the representation of the placement vector x in the orthonormal equatorial frame
of reference {e 1 , e 2 , e 3 O} at the origin O as well as in the oblique frame of reference {e 1 0 , e 2 0 , e 3 0 O}
(which is called meta-equatorial) at the origin O in terms of spherical coordinates {λ, φ, r} as well as
in meta-spherical coordinates {α, β, r}, is solved by forward and backward transformations. This is
outlined in Boxes 3.3 and 3.4. For geometrical details, consult again Figs. 3.6 and 3.7. Here, we meet
the particular problem to parallel transport the oblique frame of reference {e 1 0 , e 2 0 , e 3 0 x(λ 0 , φ 0 , r)}
(which is defined at the point x(λ 0 , φ 0 , r)) in the Euclidean sense from (λ 0 , φ 0 , r) to the origin O in
order to generate the centric frame of reference {e 1 0 , e 2 0 , e 3 0 O}.
End of Solution (the second problem).
3-3 The oblique frame of reference of the sphere
The oblique frame of reference of the sphere (three Killing vectors): normal, oblique, and transverse
aspects, Killing symmetry, designs of an oblique frame of reference of the sphere.
Let us confront you here with three aspects of the sphere, which are called normal, oblique, and
transverse. These aspects form the basis of spherical coordinates of the sphere, taking into account
the three Killing vectors of symmetry, typical for a spherical surface. The first oblique frame of
reference of S
2
r is based upon the meta-North Pole, with the spherical coordinates {λ 0 , φ 0 } as design
elements. Alternatively, the second oblique frame of reference of S
2
R refers to the centric oblique plane
P
2
O , which intersects the sphere S
2
R and passes the origin O. The oblique plane P
2
O intersects S
2
R in a
so-called circular oblique equator, also called meta-equator, which is oriented by two Kepler elements
{Ω, I}, namely the longitude Ω of the ascending node and the inclination I. Finally, the third frame
of reference, which is called transverse, is defined as special oblique, namely by an inclination I = π/2.
For all three frames of reference, we present to you the forward and backward transformation formulae,
also called direct and inverse. Their derivation is technically done in a way which is suitable for other
figures of reference which have less Killing symmetry, like the ellipsoid-of-revolution.
3-31 A first design of an oblique frame of reference of the sphere
The first design of an oblique frame of reference of the sphere S
2
r is taking reference to the following
design aspects. (i) We make a choice about the three spherical coordinates {λ 0 , φ 0 , r} of the “new
North Pole” (which is also called meta-North Pole) relative to the conventional equatorial frame of
reference. We attach to the direction of the new North Pole a new equatorial frame of reference, which
is called meta-equatorial, namely {e 1 0 , e 2 0 , e 3 0 x 0 }, a set of orthonormal base vectors at the point
x(λ 0 , Φ 0 , r) =: x 0 , such that e 3 0 := x 0 / x 0 . We connect the conventional equatorial frame of
reference {e 1 , e 2 , e 3 O} to the oblique or meta-equatorial frame of reference {e 1 0 , e 2 0 , e 3 0 O} at the
origin O, namely by parallel transport of {e 1 0 , e 2 0 , e 3 0 } from x 0 to P O (in the Euclidean sense). (ii)
We represent the coordinates of the placement vector x, both in the conventional equatorial frame of
reference and in the oblique equatorial frame of reference. (iii) We finally derive the forward as well
as backward equations of transformation between them.
Solution (the first problem).
The first problem can be solved (i) by representing the placement vector x(λ 0 , φ 0 , r) in terms
of the chosen spherical coordinates {λ 0 , φ 0 , r}, (ii) by computing the triplet of partial derivatives
{D λ 0 x, D φ 0 x, D r x}, which are normalized by the Euclidean norms D λ 0 x , D φ 0 x , and D r x ,
leading to the triplet −e φ 0 := D φ 0 x/ D φ 0 x , +e λ 0 := D λ 0 x/ D λ 0 x , and +e r := D r x/ D r x ,
and these three triplet terms are called South, East, and Vertical. Finally, we relate the base vectors
{e 1 0 , e 2 0 , e 3 0 x 0 } := {e λ 0 , e φ 0 , e r x 0 } to {e 1 , e 2 , e 3 O}. This is outlined in Box 3.3. For geometrical
details, consult Figs. 3.6 and 3.7.
End of Solution (the first problem).
Solution (the second problem).
The second problem, the representation of the placement vector x in the orthonormal equatorial frame
of reference {e 1 , e 2 , e 3 O} at the origin O as well as in the oblique frame of reference {e 1 0 , e 2 0 , e 3 0 O}
(which is called meta-equatorial) at the origin O in terms of spherical coordinates {λ, φ, r} as well as
in meta-spherical coordinates {α, β, r}, is solved by forward and backward transformations. This is
outlined in Boxes 3.3 and 3.4. For geometrical details, consult again Figs. 3.6 and 3.7. Here, we meet
the particular problem to parallel transport the oblique frame of reference {e 1 0 , e 2 0 , e 3 0 x(λ 0 , φ 0 , r)}
(which is defined at the point x(λ 0 , φ 0 , r)) in the Euclidean sense from (λ 0 , φ 0 , r) to the origin O in
order to generate the centric frame of reference {e 1 0 , e 2 0 , e 3 0 O}.
End of Solution (the second problem).