Geodetic Mapping - Map Projections: Cartographic Information Systems - page 472

Corollary 20.1 ( κ g ,κ n ,τ g ).

U M 1 + U K 1 U L 1 M 1 U M 2 + U K 2 U L 2 M 2 ,

κ g = G M 1 M 2

(20.28)

K 1 L 1 K 2 L 2

κ n = H KL U K U L ,

(20.29)

τ g = H KL U K U L + N

H NM U K U L U M

×

KL

G M 1 M 2 U M 1 + U K 1 U L 1 M 1 U M 2 + U K 2 U L 2 M 2 − 1 / 2

× . (20.30)

K 1 L 1 K 2 L 2

End of Corollary.

The curve C is called geodesic if κ g =0andthecurve C is called a geodesic circle if κ g =
const., κ n = const., and τ g = 0. Compare with Examples 20.1 and 20.2 .

Example 20.1 (Geodesic as a submanifold in { E

A 1 ,A 2 ,G KL },L = c = const: “meridian”).

A 1 cos T cos c A 1 cos T sin c

+ E 3 A 1 (1 − E 2 )sin T

X =+ E 1 1

+ E 2

1 1 ,

(20.31)

E 2 sin 2 T E 2 sin 2 T E 2 sin 2 T

− − −

X / X

D 1 = − E 1 sin T cos c − E 2 sin T sin c + E 3 cos T =: ,

D 2 = E 1 sin c − E 2 cos c, (20.32)

D 3 = E 1 cos T cos c + E 2 cos T sin c + E 3 sin T,

⎡ ⎤

− sin T cos c − sin T sin c cos T
sin c − cos c 0
cos T cos c cos T sin c sin T

⎣ ⎦ E ,

D = (20.33)

D =R E , D =R E =R R T D =Ω D ∀ R ∈ SO(3) ,

(20.34)

⎡ ⎤ ⎡ ⎤

T cos T cos c T cos T sin c T sin T T

000
+ T 00

− − − 00 −

⎣ ⎦ , Ω:=R R T = ⎣ ⎦ ,

R =

0 0 0 (20.35)

T sin T cos c T sin T sin c + T cos T − −

κ g =0 ,κ n = −T ,τ g =0 ,

E 2 ) E 2 sin 2 T ) 3 / 2

A 1 (1

= dS

(1 − E 2 sin 2 T ) 3 / 2 , T := d T

A 1 (1 −

d S = (1

−

X

dT =

, (20.36)

− E 2 )

E 2 sin 2 T ) 3 / 2
A 1 (1

(1 −

κ g =0 , n = − ,τ g =0 . (20.37)

− E 2 )

End of Example.

Obviously, the meridian L =const.isa geodesic .