Geodesics - Map Projections: Cartographic Information Systems - page 719

D 1 = G K,L U K U L + G K U K , (E.1)

D 3 = G 3 = G 3 ,L U L .

(E.2)

The Gauss-Weingarten derivational equations govern surface geometry, in particular

G K,L = M

G M + H KL G 3

( Gauss 1827 ) ,

KL

H LM G MK G K G 3 ,L = − ( Weingarten 1861 ) . (E.3)

Once being implemented into D 1 and D 3 , we gain ( E.4 ), to be confronted with ( E.5 ).

D 1 = M

U K U L G M + U K G K + H KL U K U L D 3 ,

(E.4)

KL

D 3 = −H LM G MK G K ,
D 1 =+ κ g D 2 + κ n D 3 ,

(E.5)

D 3 =

− κ n D 1 − τ g D 2 .

Proof (Corollary 20.1 ).

First statement:

M
KL

U K U L + U M G M = κ g D 2

= ⇒

κ g =

κ g D 2 | κ g D 2 = (E.6)

= M 1

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

K 2 L 2

U K 2 U L 2 + U M 2

K 1 L 1

⇒
( 20.28 ) .
Second statement:
=

=

⇒
κ n = H KL U K U L

(E.7)

⇒
( 20.29 ) .

=

Third statement:
D 1 = κ g D 2 + κ g D 2 + κ n D 3 + κ n D 3 ,
D 2 = − κ g D 1 + τ g D 3 (E.8)