20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates 417
Section 20-4.
Section 20-4 focuses on the celebrated Fermi coordinates which extend the notion of a geodetic projection. An initial point {L 0 , B 0 } is chosen. A moving point {L, B} is projected at right angles onto
the point P F , which is fixed. The two-step solution from {L 0 , B 0 , u, v, u F , v F } to {L, B, A
c
P F } is given.
{u, v} are the Fermi coordinates of the moving point and {u F , v F } are the Fermi coordinates of the
geodetic projections {L, B} onto {u F , v F }. The geodetic projections are fixed by identifying the coordinates of the point {u F , v F }.
Section 20-5.
Section 20-5 reviews all the details of Riemann coordinates, compares them with the Soldner coordinates, and additionally compares them with the Gauss–Krueger coordinates. First, we introduce
the left deformation analysis or distortion analysis of the Riemann mapping, namely by outlining the
additive measure of deformation, called the left Cauchy–Green deformation tensor. Solving the general eigenvalue/eigenvector problem for the pair {C l , G l } of symmetric matrices, G l positive-definite,
we succeed to compute and illustrate the principal distortions of the Riemann mapping. We conclude
with a global distortion analysis generated by charting the ellipsoid-of-revolution E
2
A 1 ,A 2
by means
of conformal Gauss–Krueger coordinates, parallel Soldner coordinates and normal Riemann coordinates summarized by the Airy measure of total deformation or total distortion for a symmetric strip
[−l E , +l E ] × [−b N , +b N ] relative to a point {L 0 , B 0 }. A special highlight is Table 20.5, comparing those
three coordinate systems in favor of normal Riemann coordinates. Assume a celestial body like the
Earth can be globally modeled by an ellipsoid-of-revolution. Then in terms of the Airy measure of total deformation on a symmetric strip [−l E , +l E ] × [−b N , +b N ] given by Table 20.5, normal coordinates
produce the minimal Airy global distortion when compared to parallel Soldner coordinates and conformal Gauss–Krueger coordinates. Let us therefore push forward the geodetic application of normal
Riemann coordinates.
20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates
Riemann polar/normal coordinates. Frobenius matrix, Gauss matrix, Hesse matrix, Christoffel symbols.
Surface fundamental forms.
Let there be given the ellipsoid-of-revolution E
2
A 1 ,A 2
(biaxial ellipsoid, spheroid, with semi-major A 1 ,
with semi-minor axis A 2 , and with relative eccentricity E
2 := (A
2
1 − A
2
2 )/A
2
1 ). It is embedded in
E
3 := {R
3 , δ IJ }, the three-dimensional Euclidean space of canonical metric I = {δ IJ } of Kronecker
type. The Latin indices I and J are elements of {1, 2, 3}.
E
2
A 1 ,A 2
:= {X ∈ R
3
|(X
2 + Y
2 )/A
2
1 + Z
2 /A
2
2 = 1} .
(20.1)
The ellipsoid-of-revolution E
2
A 1 ,A 2
is globally covered by the union of the two charts {L, B} and {U, V }
constituted by {ellipsoidal longitude, ellipsoidal latitude} and {meta-longitude, meta-latitude}, in
particular (20.2), for open sets (20.3) illustrated in Fig. 20.2.
X I = +E 1
A 1 cos B cos L
√
1−E 2 sin 2 B
+E 2
A 1 cos B sin L
√
1−E 2 sin 2 B
+E 3
A 1 (1−E
2 ) sin B
√
1−E 2 sin 2 B
versus
X II = +E 1
A 1 cos V cos U
√
1−E 2 sin 2 U cos 2 V
+E 2
A 1 (1−E
2 ) cos V sin U
√
1−E 2 sin 2 U cos 2 V
+
+E 3
A 1 sin V
√
1−E 2 sin 2 U cos 2 V
,
(20.2)
0 < L < 2π , −
π
2 < B < +
π
2 ,
0 < U < 2π , −
π
2 < V < +
π
2 .
(20.3)
Section 20-4.
Section 20-4 focuses on the celebrated Fermi coordinates which extend the notion of a geodetic projection. An initial point {L 0 , B 0 } is chosen. A moving point {L, B} is projected at right angles onto
the point P F , which is fixed. The two-step solution from {L 0 , B 0 , u, v, u F , v F } to {L, B, A
c
P F } is given.
{u, v} are the Fermi coordinates of the moving point and {u F , v F } are the Fermi coordinates of the
geodetic projections {L, B} onto {u F , v F }. The geodetic projections are fixed by identifying the coordinates of the point {u F , v F }.
Section 20-5.
Section 20-5 reviews all the details of Riemann coordinates, compares them with the Soldner coordinates, and additionally compares them with the Gauss–Krueger coordinates. First, we introduce
the left deformation analysis or distortion analysis of the Riemann mapping, namely by outlining the
additive measure of deformation, called the left Cauchy–Green deformation tensor. Solving the general eigenvalue/eigenvector problem for the pair {C l , G l } of symmetric matrices, G l positive-definite,
we succeed to compute and illustrate the principal distortions of the Riemann mapping. We conclude
with a global distortion analysis generated by charting the ellipsoid-of-revolution E
2
A 1 ,A 2
by means
of conformal Gauss–Krueger coordinates, parallel Soldner coordinates and normal Riemann coordinates summarized by the Airy measure of total deformation or total distortion for a symmetric strip
[−l E , +l E ] × [−b N , +b N ] relative to a point {L 0 , B 0 }. A special highlight is Table 20.5, comparing those
three coordinate systems in favor of normal Riemann coordinates. Assume a celestial body like the
Earth can be globally modeled by an ellipsoid-of-revolution. Then in terms of the Airy measure of total deformation on a symmetric strip [−l E , +l E ] × [−b N , +b N ] given by Table 20.5, normal coordinates
produce the minimal Airy global distortion when compared to parallel Soldner coordinates and conformal Gauss–Krueger coordinates. Let us therefore push forward the geodetic application of normal
Riemann coordinates.
20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates
Riemann polar/normal coordinates. Frobenius matrix, Gauss matrix, Hesse matrix, Christoffel symbols.
Surface fundamental forms.
Let there be given the ellipsoid-of-revolution E
2
A 1 ,A 2
(biaxial ellipsoid, spheroid, with semi-major A 1 ,
with semi-minor axis A 2 , and with relative eccentricity E
2 := (A
2
1 − A
2
2 )/A
2
1 ). It is embedded in
E
3 := {R
3 , δ IJ }, the three-dimensional Euclidean space of canonical metric I = {δ IJ } of Kronecker
type. The Latin indices I and J are elements of {1, 2, 3}.
E
2
A 1 ,A 2
:= {X ∈ R
3
|(X
2 + Y
2 )/A
2
1 + Z
2 /A
2
2 = 1} .
(20.1)
The ellipsoid-of-revolution E
2
A 1 ,A 2
is globally covered by the union of the two charts {L, B} and {U, V }
constituted by {ellipsoidal longitude, ellipsoidal latitude} and {meta-longitude, meta-latitude}, in
particular (20.2), for open sets (20.3) illustrated in Fig. 20.2.
X I = +E 1
A 1 cos B cos L
√
1−E 2 sin 2 B
+E 2
A 1 cos B sin L
√
1−E 2 sin 2 B
+E 3
A 1 (1−E
2 ) sin B
√
1−E 2 sin 2 B
versus
X II = +E 1
A 1 cos V cos U
√
1−E 2 sin 2 U cos 2 V
+E 2
A 1 (1−E
2 ) cos V sin U
√
1−E 2 sin 2 U cos 2 V
+
+E 3
A 1 sin V
√
1−E 2 sin 2 U cos 2 V
,
(20.2)
0 < L < 2π , −
π
2 < B < +
π
2 ,
0 < U < 2π , −
π
2 < V < +
π
2 .
(20.3)