Geodetic Mapping - Map Projections: Cartographic Information Systems - page 494

Table 20.4 Fermi coordinates: input { L 0 ,B 0 ,x,y } versus output { L,B } .Parttwo

Ellipsoidal coordinates, footpoint P F :
B F =49 ◦ 36 49”.565 8, L F =11 ◦ 2 50”.4690.
Azimuth α F0 :
α F0 =49 ◦ 15 46”.4706.
Azimuth α FP :
α FP = 319 ◦ 15 46”.4706.
Azimuth α PF :
α PF = 318 ◦ 44 47”.4562.
Ellipsoidal coordinates, point P :
B = B P =49 ◦ 13 41”.779 7, L = L P =11 ◦ 43 38”.092 5.
Length of the geodesic s = P 0 −

P as a result from the corresponding
boundary value problem:
s = 243 070 . 157 m.

Azimuth of s in P 0 :
α 0 P =35 ◦ 12 9”.629 4.
Azimuth of s in P :
αP 0 =33 ◦ 9 22”.334 2.

20-5 Deformation Analysis: Riemann, Soldner, Gauss-Krueger
Coordinates

Riemann coordinates, Soldner coordinates, and Gauss-Krueger coordinates. Deformation
analysis: Cauchy-Green matrix, Jacoby matrix, principal distortions.

In the following, the metric d S 2 of

2
A 1 ,A 2

is to be compared with the metric d s 2 of the chart

E

established by Riemann normal coordinates { x, y } ,namely

d S 2 = G KL ( U M )d U K d U L versus d s 2 = g kl ( u m )d u k d u l ,

(20.132)

d S 2 = N 2 ( B )cos 2 B d L 2 + M 2 ( B )d B 2 versus d s 2 =d x 2 +d y 2 = r 2 d α 2 +d r 2 ,

(20.133)

subject to the mapping equations x ( l, b )( ( 20.82 )). By pullback ,we

derive the left Cauchy-Green tensor C l with the right metric matrix G r =I 2 and the left Jacobian
matrix J l ,namely

→ ( 20.81 )) and y( l, b )( →

∂x i
∂U K δ ij

∂x j
∂U L ,

C l =J l G r J l or c KL = (20.134)

J l = ∂x

=

∂x
∂B

∂L

∂y
∂L

∂y
∂B